You know the drill: start by examining every check. There are three to find—Nh6, Ne7, and Qg4. Start with 1. Nh6, which forces Kh8 or more likely Kg7. What check can White then play? If he tries 2. Qg4+, then Black plays KxN. So 1. Nh6+ doesn’t look very fruitful, at least on inspection of where it leads anytime soon. And Ne7+ just loses the knight. But then there is 1. Qg4+. If Black responds by moving his king, White can play 2. Qg7 and mate. Since moving the king therefore is out of the question and Black has no way to capture the threatening queen, his only remaining option is to interpose something in front of his king. He has to play Qg6. Pause to visualize the resulting position and ask whether you can do anything with it. Yes, the Black king and queen are now arranged to be forked by the White knight via Ne7+.

Another train of thought leading to the same outcome might start by observing that White’s knight on f5 attacks the g7 square in front of Black’s king, which is exposed; this suggests the possibility of mate if White can get his queen onto g7. The natural route to that result starts with Qg4, putting the queen on the right file. Black’s response is forced: Qg6. Then you see the fork.