White is down a rook and subject to a pin: Black’s queen pins the rook on b5 to the queen behind it. Of course you also notice the concentration of activity on the seventh rank, where if White’s knight were to move, White’s queen would pin Black’s bishop. But the immediate issue is the fate of the rook on b5. If it leaves its square its queen gets taken, so its reason for doing so had better be to give check or join in a mating attack. Does White currently attack any squares adjacent to Black’s king that might serve as the basis for a mate threat?
Yes: the bishop on b2 attacks the bishop on g7, and the knight on f7 attacks h6 and h8. The rook can add another attack to h6, too, with Rh5. If Black takes the rook with his g6 pawn, White obviously has QxQ; less obviously but more importantly, White has Qf5#. But that doesn't settle the issue; the important question is what White's rook would be able to do from h5 after Black plays QxQ. The first follow-up to consider—checking with the rook via Rxh6+—doesn't work; it results in Black simply playing BxR. Yet with three White pieces then surrounding Black's king, and the king’s movements highly constrained, one must press hard. Look for another check White could give once the rook is on h5 and Black has spent a move playing out the pin with QxQ.
All right, did you see it? White has a checking option besides Rxh6+; he has Ng5+. Black's pawn on h6 would be unable to take the knight because it would be pinned by the rook. Instead Black would have to move his king to h8. Every time the king moves you ask whether new pins have been created; when the king moves to h8 here, it creates a pin of the g7 Black bishop by White’s bishop. Black takes advantage by moving onto the square Black’s bishop formerly protected: Rxh6#. Indeed, after White’s initial play of Rh5, mate follows soon no matter what Black does.
The point is to not be afraid to think about moving a relatively pinned piece if in doing so it would be joining an attack on the enemy king. Careful analysis may reveal that the move leads to a mating attack. Think in terms not just of saving the piece but of using it.