Why does the computer like this move so much?

It's black to play. The computer says that there is one move which is substantially better than any other move (more than 0.5 pawns better than the second best move).

I don't understand this at all. It's such a quiet position. Why is this move considered so much better than any other random positional move?

A move like Bd6 allows c3. Nd5 prevents c3. c3 undermines the strong pawn on d4, so preventing c3 is the most logical plan

You could see that for every suggested move, the engine gives, apart from a numeric evaluation, a line of what it considered as best play for both sides. These line are often helpful when it comes to understanding the logic behind the suggested move.

Check the 2nd best move 14...Bd6. Stockfish says white is no longer worse after 15. c4(c3) dxc3 16. Bxc3, so you can conclude (with a grain of salt) that 14...Nd5 is good because it stops c2-c4(c3).

From a positional perspective: (in the position just before 14...Nd5) what is white going to do? It’s clear that he is in desperate need of space. Therefore he might play Nd2-e4 requesting a trade, or c2-c4 to give himself more space on the queenside. 14...Nd5 prevents both so it can’t be a bad move.

Keep in mind that White is threatening Nxe5.

...Nd5 effectively meets the Nxe5 cheapo, and also prevents c2-c3. What more than that would you wish?

Sometimes it's good to consider:

If it were my opponent's move now, what would they like to play?

The opposing bishops on h1-a8 diagonal are each dangerous when exposed. When, is the key word because of the timing. Under current position, the a8 rook is vulnerable after an exchange like: 1.Nxe5 Nxe5 2.Bxb7 Nxd3 3.Bxa8 Nxe1 4.Qxe1 Rxa8 5.Bxd4

The big center is all of a sudden gone and black is a pawn down with the queen, bishop, and rook undefended. All while the knight on f6 does nothing.

If the knight is on d5, the exchange favors black. More likely relevant now, it covers the c3 square.

Bd6...stops the exchange, but c3 or c4 is coming.

I thought it would be 14... a5 (and yeah, I missed Nxe5).

White has 3 pawn breaks. 

f4

c3

a4

The most immediate, and closest to the center is c3.  So ...Nd5 would be the move, and it also stops any ideas of white playing Bb7.

Thanks for the help, everyone!

I guess I never really saw c3 as a threat. It's not something I've trained myself to look for. But it is very useful to see a logical reason for the computer's moves.

Good question! Took me 3 tries to get the move right ;D

 

I never even considered c3! Only thing I noticed was the double attack on the e5 pawn. But the pawn is double protected by the Queen and the Nc6. No advantage to white.

I accidentally got the move correct (Nd5) only because I wanted to give the bishop on e7 some attacking lanes. 

No that is an illusion, the black queen also needs to protect the bishop on b7. Therefore the pawn on e5 is in fact not protected. So white simply threatens to win the e5 pawn. But after black's move 14... Nd5 white can no longer win the pawn without loosing a piece.

That was actually what pfren pointed out in post number 5 when he wrote "...Nd5 effectively meets the Nxe5 cheapo,".

Hint: Look at the diagonal from white's bishop on g2 to black's bishop on b7.