user3305: 
What is the thought process behind playing d5 in this position?

user3594: user3447603 has already given a nice answer, so I'll just add something orthogonal to his: Bear in mind that d5 comes with a double attack, c6 and e6, so black's ideal response of e5 is not allowed anymore (to try and keep e-file blocked). So either he takes it and destroys his center and gives up the e-file, or he moves the knight and allows black to take on e6 (not the best move but still) and play around the weakened central pawns (also f5 coming in some variations). 
Finally this position demands for crude forcing moves, simply because practically speaking you don't want to give black the opportunity to long castle and secure his king. 
On a last note, all things said, one shouldn't get carried away here and wrongly assume that any crude and aggressive attack is going to work out, after all no matter how advantageous your position, you need to calculate things.

user3895: I'm not sure if it's the best one but...
First of all, it breaks black's pawn structure. Look at black pieces: Bf8, Nh7, rooks haven't moved, King is in the center. If you have safe king, developed pieces and your oponent is miles behind you, you want to open files. d5 is a move that I would do without much thinking. Such move just must be good!
After 1. ... - exd5 2. N4c3 and pawn on d5 falls. Be6 doesn't work due to simple f5. This position is much much better for White. Look at this structure. Pawn on d6, black pieces. Argh, I don't like it soo much from black's perspective. And keep in mind, pawn on g5 is hanging because of 1. Bh7 - Rh7 2. fg5 - hg5 3. Bg5.

3305: <p><img src="https://i.stack.imgur.com/dqBn3.png" alt="enter image description here"></p>

<p>What is the thought process behind playing d5 in this position?</p>
 3594: <p>user3447603 has already given a nice answer, so I'll just add something orthogonal to his: Bear in mind that d5 comes with a double attack, c6 and e6, so black's ideal response of e5 is not allowed anymore (to try and keep e-file blocked). So either he takes it and destroys his center and gives up the e-file, or he moves the knight and allows black to take on e6 (not the best move but still) and play around the weakened central pawns (also f5 coming in some variations). </p>

<p>Finally this position demands for crude forcing moves, simply because practically speaking you don't want to give black the opportunity to long castle and secure his king. </p>

<p>On a last note, all things said, one shouldn't get carried away here and wrongly assume that any crude and aggressive attack is going to work out, after all no matter how advantageous your position, you need to calculate things.</p>
 3895: <p>I'm not sure if it's the best one but...
First of all, it breaks black's pawn structure. Look at black pieces: <code>Bf8</code>, <code>Nh7</code>, rooks haven't moved, King is in the center. If you have safe king, developed pieces and your oponent is miles behind you, you want to open files. <code>d5</code> is a move that I would do without much thinking. Such move just must be good!</p>

<p>After <code>1. ... - exd5 2. N4c3</code> and pawn on d5 falls. <code>Be6</code> doesn't work due to simple <code>f5</code>. This position is much much better for White. Look at this structure. Pawn on d6, black pieces. Argh, I don't like it soo much from black's perspective. And keep in mind, pawn on g5 is hanging because of <code>1. Bh7 - Rh7 2. fg5 - hg5 3. Bg5</code>.</p>
