user8486: I was given the below position in the US chess magazine. Author said this position is "hopeless for black". Lichess engine says 1.5 advantage for white.
Why? I understand "piece points" are only a rule of thumb, but I also don't have any other heuristics to judge why white side is better. I assume that even though white is down material, she has a "positional advantage," but I don't know how to objectively discern why that is.
Why would people say white has an advantage here?
[fen ""]

1. e4 e5 2. Nf3 Nc6 3. Bb5 g6 4. c3 a6 5. Ba4 Bg7 6. d4 b5 7. Bb3 exd4 8. cxd4 Nge7 9. d5 Na5 10. Bd2 Bxb2 11. Bxa5 Bxa1 12. Nc3 Bxc3+ 13. Bxc3

EDIT
The real question I meant to ask is this: why are all "these positional things" (space, development, castling rights, etc) worth a material sacrifice of 3 pawn points!? I know these principles are good, but how am I supposed to personally know they're that good that I should be willing to sacrifice a major piece for a minor piece?

user12449: By material, white is down three pawns. (R+P vs B)  -3
White has the bishop pair, which is ~pawn            1
White has more space ~ 1/2 pawn                      1/2
White has great lead in development ~pawn            1
Black's king stuck in the middle ~pawn               1
So the positional advantages outweigh the material deficit.
Black's best moves are f6 and Rg8.  Both concede a pawn and, most likely an exchange due to the weakness of the dark squares(Ng5-h7-f6+).  Black will need at least four moves to castle to the queenside, and that leaves him still facing pressure down the half-open files.  The most important factor is that it's more difficult to defend than to attack, which increases white's value.  These elements shows that white is easily up by at least the value of a pawn.

user70: This answers your question by showing the pull White has. White's advantage is real. But I think it is important to notice how resilient a defender can be. This is a great example of how difficult it can be to win a "won" game. I am pretty sure Stockfish could beat me from either side of the board.
The following is Stockfish 6 playing each side of the board, 5 minutes per move.

[FEN "r1bqk2r/2ppnp1p/p5p1/1p1P4/4P3/1BB2N2/P4PPP/3QK2R b Kkq - 0 1"]

1...Rf8 2.Ng5 Ng8 3.h4 d6 4.Nxh7 Qe7 5.f3 Bd7 6.Nxf8 Qxf8 7.Qc1 f6 8.Bd2 Kd8 9.Ba5 Ke8 10.Qxc7 Nh6 11.O-O Qe7 12.Qb7 Rc8 13.Qxa6 Nf7 14.Qb6 Kf8 15.Re1 Qe5 16.Qf2 g5 17.Bd2 Kg8 18.hxg5 fxg5 19.f4 gxf4 20.Bxf4 Qg7 21.e5 dxe5 22.Bh2 Ng5 23.d6+ Kh7 24.Qh4+ Qh6 25.Qxh6+ Kxh6 26.Rxe5 Nh7 27.Bf4+ Kg6 28.Kf2 Bf5 29.Ke3 Nf6 30.Kd4 b4 31.Re7 Nd7 32.Bd2 Nb8 33.Kd5 Kf6 34.Bxb4 Nd7 35.a4 Nb6+ 36.Kd4 Nd7 37.Be1 Rc1 38.a5 Rc8 39.Bc4 Nb8 40.Bc3 Kg6 41.Kd5 Kg5 42.Rg7+ Kh5 43.Rc7 Rd8 44.Be2+

...and at this point Stockfish rated the position at +20. I wanted my computer back ;-)

user11629: White has a lot of compensation for the sacrificed exchange, specifically:

more space: the white central pawns are perfectly placed to limit the movement of the black knight and bishop
play on the dark squares: black has considerably weakened the dark squares on the king side and does not have a dark-squared bishop anymore; White can easily take advantage of that by operating on these squares with bishop, knight and queen.
lead in development: Basically white only needs to activate the rook, while black really struggles to free himself. Also black will be forced to move the rook on h8 (castling seems too dangerous), losing another move for development and leaving the king in the center for some more time where it will be under attack.

This suggests that white can make use of these elements, before black can recover.
Somewhat concrete lines:
Black's rook h8 is under attack and has to move. Three options:

1...0-0 looks very dangerous for black. White could just play something like Qd2, Qh6, Ng5 and if necessary h4, h5. I don't see what black could do to avoid getting mated soon. Note that black cannot play something like f6 soon because of the discovered check from the b3 bishop.
1..Rg8 or 1..Rf8. Here white has basically two ideas, which can be combined in various ways. a) play d6 which prevents black from playing d6 himself (freeing the bishop on c8) and allows white to activate the queen (via d6-f6). Also, with the white queen on d6 and the white bishop on f6, white would basically bind four black pieces (Qd8, Ne7, Ke8, Rf8) to very passive positions. b) play Ng5, which hints at an attack on f7 and also on re-routing the knight via h7 to f6.

So overall, yes the position is pretty hopeless for black.

Regarding your general quesion (Why are these things worth three pawn points? And how do I know when they are worth that much?):
You should realize that material is only one of many factors relevant for judging  a position. Admittedly, in many games, players have somewhat symmetric positions, are developed equally well, etc and in this case, material becomes a major factor. So yes, in quiet normal positions material is often very important.
However in highly unbalanced positions like the one in your example, material count becomes less important and is easily trumped by activity. There is no strict rule to know when this is the case. In fact this is among the more difficult aspects of chess and can basically only gained by experience. I would strongly discourage you from trying to assess such unbalanced positions by summing points for "piece activity", "weak king"... Chess just does not work this way.
Computers do something like this in their evaluation function, but they do this for many (lots) positions many moves down the line (not only the current position) and then it does make sense. Furthermore computers have a more fine-grained number for "space", "piece activity", etc. 
Anyway, humans lack the power to do all this and basically need to rely on experience, intuition and (occasionally) calculating some more or less forced concrete lines.
As a very rough guideline (don't follow this strictly) you can sacrifice:

a pawn: to gain a lead in development/piece activity or central control, for at least a short amount of time, often with a potential to regain the pawn easily if needed (see e.g. Queen's gambit). Sacrificing a pawn is not a major deal as many positions are still a draw +- a pawn.
up to three pawns or an exchange: if you get a major long term positional advantage through it; often you end up with a very strong minor piece vs. a not-so strong rook (e.g. in somewhat closed positions). Here you really need to have something concrete. 
more than three pawns or an exchange: rarely, typically in direct mating attacks only


8486: <p>I was given the below position in the US chess magazine. Author said this position is "hopeless for black". Lichess engine says 1.5 advantage for white.</p>

<p>Why? I understand "piece points" are only a rule of thumb, but I also don't have any other heuristics to judge why white side is better. I assume that even though white is down material, she has a "positional advantage," but I don't know how to objectively discern why that is.</p>

<p>Why would people say white has an advantage here?</p>

<pre><code>[fen ""]

1. e4 e5 2. Nf3 Nc6 3. Bb5 g6 4. c3 a6 5. Ba4 Bg7 6. d4 b5 7. Bb3 exd4 8. cxd4 Nge7 9. d5 Na5 10. Bd2 Bxb2 11. Bxa5 Bxa1 12. Nc3 Bxc3+ 13. Bxc3
</code></pre>

<p>EDIT</p>

<p>The real question I meant to ask is this: why are all "these positional things" (space, development, castling rights, etc) worth a <em>material sacrifice</em> of 3 pawn points!? I know these principles are good, but how am I supposed to personally know they're <em>that</em> good that I should be willing to sacrifice a major piece for a minor piece?</p>
 12449: <p>By material, white is down three pawns. (R+P vs B)  -3</p>

<p>White has the bishop pair, which is ~pawn            1</p>

<p>White has more space ~ 1/2 pawn                      1/2</p>

<p>White has great lead in development ~pawn            1</p>

<p>Black's king stuck in the middle ~pawn               1</p>

<p>So the positional advantages outweigh the material deficit.</p>

<p>Black's best moves are f6 and Rg8.  Both concede a pawn and, most likely an exchange due to the weakness of the dark squares(Ng5-h7-f6+).  Black will need at least four moves to castle to the queenside, and that leaves him still facing pressure down the half-open files.  The most important factor is that it's more difficult to defend than to attack, which increases white's value.  These elements shows that white is easily up by at least the value of a pawn.</p>
 70: <p>This answers your question by showing the pull White has. White's advantage is real. But I think it is important to notice how resilient a defender can be. This is a great example of how difficult it can be to win a "won" game. I am pretty sure Stockfish could beat me from either side of the board.</p>

<p>The following is Stockfish 6 playing each side of the board, 5 minutes per move.</p>

<pre><code>
[FEN "r1bqk2r/2ppnp1p/p5p1/1p1P4/4P3/1BB2N2/P4PPP/3QK2R b Kkq - 0 1"]

1...Rf8 2.Ng5 Ng8 3.h4 d6 4.Nxh7 Qe7 5.f3 Bd7 6.Nxf8 Qxf8 7.Qc1 f6 8.Bd2 Kd8 9.Ba5 Ke8 10.Qxc7 Nh6 11.O-O Qe7 12.Qb7 Rc8 13.Qxa6 Nf7 14.Qb6 Kf8 15.Re1 Qe5 16.Qf2 g5 17.Bd2 Kg8 18.hxg5 fxg5 19.f4 gxf4 20.Bxf4 Qg7 21.e5 dxe5 22.Bh2 Ng5 23.d6+ Kh7 24.Qh4+ Qh6 25.Qxh6+ Kxh6 26.Rxe5 Nh7 27.Bf4+ Kg6 28.Kf2 Bf5 29.Ke3 Nf6 30.Kd4 b4 31.Re7 Nd7 32.Bd2 Nb8 33.Kd5 Kf6 34.Bxb4 Nd7 35.a4 Nb6+ 36.Kd4 Nd7 37.Be1 Rc1 38.a5 Rc8 39.Bc4 Nb8 40.Bc3 Kg6 41.Kd5 Kg5 42.Rg7+ Kh5 43.Rc7 Rd8 44.Be2+
</code></pre>

<p>...and at this point Stockfish rated the position at +20. I wanted my computer back ;-)</p>
 11629: <p>White has a lot of compensation for the sacrificed exchange, specifically:</p>

<ul>
<li><strong>more space</strong>: the white central pawns are perfectly placed to limit the movement of the black knight and bishop</li>
<li><strong>play on the dark squares</strong>: black has considerably weakened the dark squares on the king side and does not have a dark-squared bishop anymore; White can easily take advantage of that by operating on these squares with bishop, knight and queen.</li>
<li><strong>lead in development</strong>: Basically white only needs to activate the rook, while black really struggles to free himself. Also black will be forced to move the rook on h8 (castling seems too dangerous), losing another move for development and leaving the king in the center for some more time where it will be under attack.</li>
</ul>

<p>This suggests that white can make use of these elements, before black can recover.</p>

<p>Somewhat concrete lines:</p>

<p>Black's rook h8 is under attack and has to move. Three options:</p>

<ol>
<li>1...0-0 looks very dangerous for black. White could just play something like Qd2, Qh6, Ng5 and if necessary h4, h5. I don't see what black could do to avoid getting mated soon. Note that black cannot play something like f6 soon because of the discovered check from the b3 bishop.</li>
<li>1..Rg8 or 1..Rf8. Here white has basically two ideas, which can be combined in various ways. a) play d6 which prevents black from playing d6 himself (freeing the bishop on c8) and allows white to activate the queen (via d6-f6). Also, with the white queen on d6 and the white bishop on f6, white would basically bind four black pieces (Qd8, Ne7, Ke8, Rf8) to very passive positions. b) play Ng5, which hints at an attack on f7 and also on re-routing the knight via h7 to f6.</li>
</ol>

<p>So overall, yes the position is pretty hopeless for black.</p>

<hr>

<p>Regarding your general quesion (<em>Why are these things worth three pawn points? And how do I know when they are worth that much?</em>):</p>

<p>You should realize that material is only one of many factors relevant for judging  a position. Admittedly, in many games, players have somewhat symmetric positions, are developed equally well, etc and in this case, material becomes a major factor. So yes, in quiet normal positions material is often very important.</p>

<p>However in highly unbalanced positions like the one in your example, material count becomes less important and is easily trumped by activity. There is no strict rule to know when this is the case. In fact this is among the more difficult aspects of chess and can basically only gained by experience. I would strongly discourage you from trying to assess such unbalanced positions by summing points for "piece activity", "weak king"... Chess just does not work this way.</p>

<p>Computers do something like this in their evaluation function, but they do this for <em>many</em> (lots) positions <em>many</em> moves down the line (not only the current position) and then it does make sense. Furthermore computers have a more fine-grained number for "space", "piece activity", etc. </p>

<p>Anyway, humans lack the power to do all this and basically need to rely on experience, intuition and (occasionally) calculating some more or less forced concrete lines.</p>

<p>As a very rough guideline (don't follow this strictly) you can sacrifice:</p>

<ul>
<li><strong>a pawn</strong>: to gain a lead in development/piece activity or central control, for at least a short amount of time, often with a potential to regain the pawn easily if needed (see e.g. Queen's gambit). Sacrificing a pawn is not a major deal as many positions are still a draw +- a pawn.</li>
<li><strong>up to three pawns or an exchange</strong>: if you get a major long term positional advantage through it; often you end up with a very strong minor piece vs. a not-so strong rook (e.g. in somewhat closed positions). Here you really need to have something concrete. </li>
<li><strong>more than three pawns or an exchange</strong>: rarely, typically in direct mating attacks only</li>
</ul>
