Game analysis: What would motivate a human player to sacrifice White Knight at move 10 in this game? I am a beginner player (rating on Lichess.org around 1550). While analyzing one of my own games (pasted below), Stockfish calculated that a Knight sacrifice at 10th move is a good idea. I tried playing through the position after the Knight sacrifice, and white is able to regain a minor piece after ~10 moves, and it possibly worsens Black's position after ~10 more moves (but I'm not so sure about that). 

But, my question is: is it possible for a human player to recognize that a Knight sacrifice at move 10 in this game is going to benefit White? Is there a simple heuristic thought process that can help a player recognize that the position after 10. Nxe5 is beneficial for White?

(My chess knowledge level: trying to play without hanging pieces. No knowledge of openings or positional games or any formal theory)

The game is as follows:

[fen ""]
[White "White"]
[WhiteElo "1585"]
[Black "Black"]
[BlackElo "1354"]

1. e4 e5 2. Nf3 Qf6 { C40 King's Pawn Game: McConnell Defense } 3. Nc3 c6 4. Bc4 h6 5. O-O Be7 6. d4 d6 7. d5 c5 8. Nb5 Na6 9. c3 Qg6 { I am surprised that Stockfish thinks a Knight sacrifice Nxe5 is better here. Let me see how that line goes.... } 10. Re1 (10. Nxe5 dxe5 11. d6 Bh3 (11... Bd8 12. d7+ Bxd7 13. Nd6+ Kf8 14. Nxf7 Bc6 15. Nxd8 Qxe4 16. Nxc6 Qxc6 17. Qh5 Qc7 18. f4 e4 19. f5 Nf6 20. Qg6 b5 21. Bf4 Qd7 22. Bd6+ Qxd6 23. Qf7#) 12. Qf3 Be6 13. Bxe6 Qxe6 14. dxe7 Nxe7 15. Rd1 O-O 16. Rd6 Qc4 17. Na3 Qa4 18. b3 Qe8 19. Nc4 Ng6 20. Be3 Rd8) 10... Bh3 11. Bf1 Bf6 12. Qa4 Bd7 13. b4 c4 14. Bxc4 Ne7 15. Nxd6+ Kd8 16. Nxb7+ Kc8 17. Qxa6 Bg5 18. Nd6+ Kd8 { Here White missed a mate in 1 with Qa5 } 19. Nxe5 Qf6 20. Qb7 Qxd6 21. Nxf7+ Ke8 22. Nxd6+ { Black resigns. } 1-0
I am quite certain that it is possible to find this move for a higher level chess player. 

This has little to do with calculation depth actually. Yes, the engine is able to look 10 moves ahead and conclude that White is going to win his piece back in most or all positions resulting from the sacrifice. But for a human player, all that matters is if he can "see" (recognize) the right pattern. Concrete calculation only follows to make sure that this pattern is actually winning (and most of the work is done in the first 3 moves, instead of 10). Unfortunately, there is no easy heuristic blueprint (else we would all be GMs), most of this ability to "see" comes from experience. However, there are a few hints:


First of all, White has the more active position (he has castled already, his pieces are much better developed (to active squares), and he has some space advantage), so he should generally put himself in the mindset of the attacker, not the defender.
Looking at the developed pieces in particular, the knight on b5 is very strong and puts pressure on c7 and d6 - and from the latter could attack f7 as well - which also happens to be in the scope of his bishop on c4. 
His queen and other bishop are ready to attack as well, but for now, there are pawns in the way. A friendly one blocks the file for the queen, and a hostile one defends f4 from the bishop. Nxe5 will help in both cases, but that alone does of course not justify a sacrifice.
Now the crucial part is to switch focus from attacking d6 to occupying d6. Important features that make this pawn advance really strong all by themselves (aside from the fact that passed pawns on the 6th rank are rarely a bad idea): Black's bishop on e7 will be attacked immediately, and Black has not castled yet, so d7+ is a very serious threat. 
Additionally, all of White's pieces gain activity: The bishop on c4 gets freed, the queen gets additional scope along the d-file, the knight on b5 gets additional safety on c7 and without a black pawn on d6, the ones on c5 and e5 become very vulnerable to the other knight and bishop.


So, White's position would be overwhelmingly strong - if only the black pawn on d6 did not exist. A GM will probably see this immediately. For us normal folks, maybe that is the mental exercise you are looking for: If you "sense" that you have an advantageous position that possibly allows some tactics, check the important pieces and pawns (not the one on a6 obviously) of your opponent and judge the position with them missing. Once you have found a keystone, the actual calculation often is not that difficult. You just have to find a way to deflect the pawn/piece you identified as holding your opponent's position together, leading you to look out for moves like Nxe5.

Now in this case, the real calculation would merely amount to:


Bh3 counter attack - easily defended by Qf3, even adding pressure on f7 at the same time
Bxd6 - you can simply retake, regaining your piece while maintaining a good attack
Black retreats his bishop (e.g Bd8) - you can continue with d7+, and after Bxd7, d6 is open for the knight. At this point, you may have to calculate 2-3 additional moves, but really, if you have arrived here, you have done most of the work already. 
Additionally, Tony Ennis' idea around f4 is worth having a look at, too.
Last but not least, you always have to consider that Black can refuse the sacrifice. Well, then Nxe5 either just won a pawn (probably the safer choice by Black) or has opened another attack angle after ...Qxe4 - you can now play Re1 with pressure along the opened e-file on an uncastled king.


As a contrast, list of reasons why people tend to overlook opportunities like this:


They are too occupied with the last move of Black (Qg6) that threatens Qxe4 and focus solely on the defense of that threat. End of list.


In all honesty, I still belong to that group too. Hindsight is 20/20.