user12502: Usually chess puzzles at my level (beginner) have just one or two forced variations and a clear achievement, like a substantial material gain or checkmate. Once in a while, though, a puzzle leaves me wondering, why the solution is actually supposed to be such a good thing.
The following puzzle is from Susan Polgar's and Paul Truong's book "Chess Tactics for Champions", #11 from the chapter on trapping. White to move:
[White "Puzzle by Susan Polgar"]
[Black "Trapping: #11"]
[Result "*"]
[FEN "2rr2k1/p3bpp1/1p2pn1p/2p5/8/2P1PPB1/PP2B1PP/R4RK1 w - - 0 1"]

The solution is:

This example demonstrates well the power of the pair of bishops 1.♗a6 ♖c6 (1...♖a8 2.♗b7) 2.♗b7.

After 2...♖cd6 3.♗xd6 ♗xd6 (or 3...♖xd6?) White wins the exchange. So far, so good, numerical material gain. However, I'm trying to make my very first baby steps in understanding strategy: In an actual game, assuming I'd see the variation, I'd be hesitating to play it. My -- probably naive -- reasoning would go like this: "A rook is worth five pawns, a bishop three. But White has the bishop pair. Moreover, the rook on c8 doesn't do much, while the dark squared bishop has a nice diagonal. So, I'm not convinced that it's really 5 vs 3."
It's not that I don't trust Polgar/Truong, but I would like to understand. Is it, because after the exchange White is now able to control the d-file? Or am I completely off the track somewhere else?

user2789: 
A rook is worth five pawns, a bishop three. But White has the bishop pair. Moreover, the rook on c8 doesn't do much, while the dark squared bishop has a nice diagonal. So, I'm not convinced that it's really 5 vs 3.

Sure, but the rook doesn't have to stay on c8. After, say, ...Rd2 and ...Rcd8, Black's rooks start to look a lot more useful. Being up the exchange is a long-term advantage; Black's rook not being very useful on c8 is a short-term feature of the present position.

user12699: Material trumps positional factors, unless the positional factors are extremely pronounced.
The advantage of the bishop pair is often estimated at half a pawn - enough to pressure the opponent, but not always force a win. Rook vs. knight is an advantage of roughly two pawns - clearly winning - so you're gaining a lot in terms of material.
You should cash in the bishop for the rook and look forward to a clearly winning position. You're not likely to have a better opportunity.

user-1: 
A rook is worth five pawns, a bishop three. But White has the bishop pair. Moreover, the rook on c8 doesn't do much, while the dark squared bishop has a nice diagonal. So, I'm not convinced that it's really 5 vs 3.

Congratulations! You are taking your first steps to moving away from using the Reinfeld piece values as a method of evaluation.
The reason why winning the exchange is good here is because:

Material appreciates as the endgame approaches. For example, if you have 8 pawns vs. 7, you are up a pawn. If you reduce that to 1 pawn to 0, you are still up a pawn but now you are very likely to win (or certain to draw)
Building from point 1, white can take control of the crucial d-file and likely trade down more material, going into an even better endgame
Owning the bishop pair is usually worth 1/2 a pawn, but in this position the rooks are the most important pieces because the pawn structure means that the black minor pieces don't have great squares (a somewhat blocked bishop and a knight with no forward outpost)

Please note that winning material isn't always advantageous. This example in Mastering Chess Strategy by GM Hellsten shows white sacrifice the exchange to gain a good outpost for his knight and play against a bad bishop:
[FEN "r4r2/1pp1q1bk/3p2b1/p2Ppp2/1nP5/1PN1BP2/P2N2PP/2RQ1RK1 w - - 0 1"]
[White "Hellsten"]
[Black "Cramling"]

  1. Qe2            f4            
  2. Bf2            Bd3           
  3. Qd1            Bxf1          
  4. Qxf1           Bf6           
  5. Kh1            Bh4           
  6. Bg1            Rf7           
  7. Nde4           Rg8           
  8. a3             Na6           
  9. Rc2            Rg6           
 10. Rb2            Qd8           
 11. b4             axb4          
 12. axb4           Be7           
 13. c5             Nb8           
 14. Qc4            Kh8           
 15. b5             Nd7           
 16. c6             Nb8           
 17. cxb7           Rg8           
 18. Ra2            Bh4           
 19. Ra8            Rfg7          
 20. Qf1            Qd7           
 21. Ba7             1-0


Addendum: Regarding how material appreciates as the ending nears, consider the case of K+8P vs. K+7P:
[FEN "4k3/pppp1ppp/8/8/8/8/PPPPPPPP/4K3 w - - 0 1"]

Compared to K+P vs. K, which is the same material difference of 1 pawn but a higher material ratio for the attacker, the first example is a likely draw, whereas the second example is a theoretical win.
[FEN "4k3/8/8/8/8/8/4P3/4K3 w - - 0 1"]


12502: <p>Usually chess puzzles at my level (beginner) have just one or two forced variations and a clear achievement, like a substantial material gain or checkmate. Once in a while, though, a puzzle leaves me wondering, why the solution is actually supposed to be such a good thing.</p>

<p>The following puzzle is from Susan Polgar's and Paul Truong's book "Chess Tactics for Champions", #11 from the chapter on trapping. White to move:</p>

<pre><code>[White "Puzzle by Susan Polgar"]
[Black "Trapping: #11"]
[Result "*"]
[FEN "2rr2k1/p3bpp1/1p2pn1p/2p5/8/2P1PPB1/PP2B1PP/R4RK1 w - - 0 1"]
</code></pre>

<p>The solution is:</p>

<blockquote>
  <p>This example demonstrates well the power of the pair of bishops 1.♗a6 ♖c6 (1...♖a8 2.♗b7) 2.♗b7.</p>
</blockquote>

<p>After 2...♖cd6 3.♗xd6 ♗xd6 (or 3...♖xd6?) White wins the exchange. So far, so good, numerical material gain. However, I'm trying to make my very first baby steps in understanding strategy: In an actual game, assuming I'd see the variation, I'd be hesitating to play it. My -- probably naive -- reasoning would go like this: "A rook is worth five pawns, a bishop three. But White has the bishop pair. Moreover, the rook on c8 doesn't do much, while the dark squared bishop has a nice diagonal. So, I'm not convinced that it's really 5 vs 3."</p>

<p>It's not that I don't trust Polgar/Truong, but I would like to understand. Is it, because after the exchange White is now able to control the d-file? Or am I completely off the track somewhere else?</p>
 2789: <blockquote>
  <p>A rook is worth five pawns, a bishop three. But White has the bishop pair. Moreover, the rook on c8 doesn't do much, while the dark squared bishop has a nice diagonal. So, I'm not convinced that it's really 5 vs 3.</p>
</blockquote>

<p>Sure, but the rook doesn't have to stay on c8. After, say, ...Rd2 and ...Rcd8, Black's rooks start to look a lot more useful. Being up the exchange is a long-term advantage; Black's rook not being very useful on c8 is a short-term feature of the present position.</p>
 12699: <p>Material trumps positional factors, unless the positional factors are <em>extremely</em> pronounced.</p>

<p>The advantage of the bishop pair is often estimated at half a pawn - enough to pressure the opponent, but not always force a win. Rook vs. knight is an advantage of roughly two pawns - clearly winning - so you're gaining a lot in terms of material.</p>

<p>You should cash in the bishop for the rook and look forward to a clearly winning position. You're not likely to have a better opportunity.</p>
 -1: <blockquote>
  <p>A rook is worth five pawns, a bishop three. But White has the bishop pair. Moreover, the rook on c8 doesn't do much, while the dark squared bishop has a nice diagonal. So, I'm not convinced that it's really 5 vs 3.</p>
</blockquote>

<p>Congratulations! You are taking your first steps to moving away from using the Reinfeld piece values as a method of evaluation.</p>

<p>The reason why winning the exchange is good here is because:</p>

<ol>
<li>Material appreciates as the endgame approaches. For example, if you have 8 pawns vs. 7, you are up a pawn. If you reduce that to 1 pawn to 0, you are still up a pawn but now you are very likely to win (or certain to draw)</li>
<li>Building from point 1, white can take control of the crucial d-file and likely trade down more material, going into an even better endgame</li>
<li>Owning the bishop pair is <em>usually</em> worth 1/2 a pawn, but in this position the rooks are the most important pieces because the pawn structure means that the black minor pieces don't have great squares (a somewhat blocked bishop and a knight with no forward outpost)</li>
</ol>

<p>Please note that winning material isn't always advantageous. This example in <em>Mastering Chess Strategy</em> by GM Hellsten shows white sacrifice the exchange to gain a good outpost for his knight and play against a bad bishop:</p>

<pre><code>[FEN "r4r2/1pp1q1bk/3p2b1/p2Ppp2/1nP5/1PN1BP2/P2N2PP/2RQ1RK1 w - - 0 1"]
[White "Hellsten"]
[Black "Cramling"]

  1. Qe2            f4            
  2. Bf2            Bd3           
  3. Qd1            Bxf1          
  4. Qxf1           Bf6           
  5. Kh1            Bh4           
  6. Bg1            Rf7           
  7. Nde4           Rg8           
  8. a3             Na6           
  9. Rc2            Rg6           
 10. Rb2            Qd8           
 11. b4             axb4          
 12. axb4           Be7           
 13. c5             Nb8           
 14. Qc4            Kh8           
 15. b5             Nd7           
 16. c6             Nb8           
 17. cxb7           Rg8           
 18. Ra2            Bh4           
 19. Ra8            Rfg7          
 20. Qf1            Qd7           
 21. Ba7             1-0
</code></pre>

<hr>

<p><strong>Addendum:</strong> Regarding how material appreciates as the ending nears, consider the case of K+8P vs. K+7P:</p>

<pre><code>[FEN "4k3/pppp1ppp/8/8/8/8/PPPPPPPP/4K3 w - - 0 1"]
</code></pre>

<p>Compared to K+P vs. K, which is the same material difference of 1 pawn but a higher material ratio for the attacker, the first example is a likely draw, whereas the second example is a theoretical win.</p>

<pre><code>[FEN "4k3/8/8/8/8/8/4P3/4K3 w - - 0 1"]
</code></pre>
