Why is 9. a4 the best move? What's behind that move?

Can someone point me to the right direction please?

Your development is mostly complete. Since your queen is a bit trapped, moving it won't help much. This is the time to look into how to break through black's position and look at his weaknesses. It looks like we can put pressure on blacks queenside, since the pawn structure isn't perfect. b5 looks like a fun square for a knight to hop into, right?

9. a4 is a niche move that prepares Nb5 in the next move. If black doesn't react with something like a6, that knight on b5 might become a nuisance for black in the future. Attacks critical squares in black's camp such as d6 and c7. It also comes with a discovered attack on black's queen with your dark squared bishop.

First of all....who or what said that was the "best" move?  because i doubt it is.  But to answer your question.  I would imagine its because 9.a4 gains space, and prevents black from playing ...b5.

The firs thing i see is 9.Nd5

I'd have thought it'd be more logical to play a4 on the previous move instead of Bd2 - that way there's also the option to play the typical Nf3-d2-c4.

As it is, it still looks decent, but I'm sure many moves are good, Nd5 among other things.

Actually the d5 square is weak and light square in general for blacks. Like Nh4-f5 and Nc3-d5 and pushing f2-f4 are good way to put pressure on blacks' side. a2-a4 prevent counter attack on the queen side with b7-b5 and also let the queen been developped by Qd1-b1-a2. A typical manoeuver we can see in some lines of Italian game for example 1.e4; e5 2.Nf3; Nc6 3.Bc4; Bc5 4.c3; Nf6 5.d3; O-O 6.O-O; d6 7.Re1; h6 8.a4; a5 9.Nbd2; Be6 10.Bb5; Qb8! (◇ preparing Qa7).

Another example 1.Nc3; d5 2.e4; d4 3.Ne2; e5 4.Ng3; g6 5.Nf3; Bd6 6.Bc4; Nc6 7.a3; h5 8.h4; Bg4 9.d3; Qd7 10.Bd2; O-O-O 11.Qb1! (◇ preparing Qa2 or b2-b4-b5).

I would say instead of playing Nd5 immediately, you can first play a4 then try to bring both your knights to control the d5 square. I think that might be a little too slow but that can be a long term plan.

My idea was to bring both knights to control the square

Thanks!

This has been a useful discussion I'm sure, but when you are using the computer by yourself, focus on stuff the engine calls "blunders" (which is the big stuff) and you can decide whether to look at stuff the engine calls a "mistake" on a case by case basis (that is the intermediate stuff).  Ignore everything else.  Too often the other stuff is just not possible for most humans to parse and learn from, and that is particularly true at your current rating level.

Sure!  And when there are many stronger players including an IM willing to help, why not?  Just saying, that sometimes the computers moves are mysterious or "computery".  If you give stockfish a little longer, Pfren's nh4 is preferred.  If you follow that line, the engine seems to like g4.  Which is weird!  Sometimes even IM-level folks look at moves like that and have nothing to say other than "ok, I guess in this specific situation maybe it makes a kind of computer sense, but it is not a very human move.  I'd probably be looking at . . . "   That is why in terms of bang for your buck, its better to spend time on the blunders and sometimes also (some of) the mistakes.  

Nd5 wins a queen or a rook, that should be best. the queen's only good square is qa4, which is followed up by nc7+. i dont know why stockfish suggests a4

Last time I checked the rules, Queens could move backwards. So, 9.Nd5 Qd8 suggests itself.

Why is this topic still running after 4 years?

Besides, the OP surely misinterpreted something, a4 is not the engine's top choice as I look at it.