What should I be thinking in this position?

I'm no expert here, but just looking through your game, the chain of events that lead up to you winning included the 11. nf5 blunder which basically threw a knight in the bin for nothing. I'm pretty sure the advice you get from the engine is based on the next move also being the strongest one.
It looks to me (again, I'm a beginner as well so not to be trusted) like 11. g3 would have kicked your queen away without much of an attack left for you.

Well for one, I wouldn't attack the Q with the N, I would instead have taken the free pawn on d5. Bxd5. This changes the entire sequence of moves (I don't see the Q as an immediate threat to the light pieces). Just because this opponent attacked with the N doesn't mean a stronger player would. Threats alone cannot "force" any moves. Ne5 centralizes the N and allows for a much stronger game for the dark pieces. Face it, Q x B on f5 is a blunder, even g3 there would have given the dark pieces a better game.
Still, that is a sweet win, congrats!

Took me a moment as well....my conclusion is that he was playing black and just made a typo with "attack the black king"? Could be wrong though...