You see that White has a standard formation for a mate: his queen is ready to land next to Black’s king on g7 with protection from his knight. Black’s queen prevents this, so White considers what else the Black queen is trying to protect—and finds the rook on g5. White can take it with RxR, but this won’t quite work; once Black recaptures with QxR, his queen still guards g7. What White needs is a capture of a piece that Black can't defend without pulling his queen out of reach of g7. The queen guards nothing else, but notice that the pattern might be completed if Black’s rook could be forced onto a different square. Or if you prefer, simply consider any checks White can give. By either route you find Rh4+.
How is Black to reply? White’s rook cannot be captured, so Black’s only choices are to interpose something or move the king. Consider his principal alternatives.
(a) Black can move his king to g8. When the enemy king moves, ask what checks you would have against it on its new square; be especially alert to moves by the king that put it on the same color square as your knight if you have one in the neighborhood. Here Kg8 is met with Ne7+—a knight fork that wins the queen. Mate follows a couple of moves later.
Or (b) Black can interpose his rook at h5. But now it becomes as good as loose; if White takes it with RxR+, Black no longer can reply QxR (for White then mates with Qxg7). Yet White soon mates anyway: after White plays 2. RxR+ Black has to move his king to g8, and now comes the knight fork 3. Ne7+, etc. That sequence is why the position appears in this section. Rh4 creates a fresh threat of disaster that only Black’s queen can address, but it already has a mortal threat to worry about at g7. Once Black’s rook interposes on h5, the Black queen thus is overworked in a straightforward sense. It guards a mating square and a rook that both are under attack, and if it recaptures in one place it abandons the defense of the other.