This is an interesting and very instructive knight endgame. It highlights several themes you will find useful in your own games. First, remember Botvinnik's Rule. If the knights were gone, how would White play this position?
1.a5! bxa5 2.a6! and White's b-pawn is the first to queen. This is a basic pawn breakthrough. Moves like 1.Kxb6? are too slow to win a race.
Returning to our position with knights on-board, we would like to do the same thing, but Black's pesky knight on c4 prevents a5. We would like to trade knights, but we can't force it.
So we use a common tool in knight endings--the deflective knight sacrifice!
[Position after 1.Nd2!? Nxd2]
Before you play a crazy sacrifice, calculate as deeply as you can:
2.a5 bxa5 3.b6 Nc4 4.b7 Ne5. And not 5.b8=Q? Nc6+! -/+, but instead shouldering out the knight with the king should lead to = or +/-.
2.a5 bxa5 3.b6! a4? or 3...h4? +/- ... White's pawn is closer to the finish like and can promote before the other pawns.
2.a5 Nc4!? 3.a6 Nd6 4.Kxb6 should be = or +/-.
1...Ne5?! Kxb6 +/-
You may be able to calculate more or less deeply, or fewer or more variations, but the point is to calculate as deeply as you can. If White doesn't break through soon, he'll lose anyway, so a calculated gamble makes sense.
[Position after 2.a5 bxa5 3.b6 Nf3 4.b7 Ne5]
5.b8=Q?? Nc6+! obviously loses.
5.Ka8 Nc6! 6.b8=Q Nxb8 loses.
5.Ka6 Nc6! 6.Kb6 Nb8 and general rules tell us Black can at least force a draw, so we should pursue other lines.
5.Kb6 Nd7+ 6.Kc7 (to prevent Nb8) 7.Nc5 b8=Q 8.Na6+! and Black wins. If on the other hand White allows 6...Nb8, Black can at least force a draw, so we should pursue other lines.
6.Kb8! Nc6+ 7.Kc7 Nb4 8.Kb6 +/-
6.Kb8! Nd7+ 7.Kc8 Nb6 8.Kc7 Nc4 9.b8=Q +/-
6.Kb8! a4 7.Kc7 a3 8.b8=Q +/-
[Position after 6.Kb8! Nd7+ 7.Kc8 Nb6 8.Kc7 Nc4 9.b8=Q +/-]
A winning position.
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