Here we find another constrained king. White has three pieces—his queen and both bishops (don’t overlook the one on c3)—aimed at squares adjacent to it, though not the same ones. You pursue a mating idea in these circumstances by thinking about checks and their results. The most powerful tool for attacking the king is your queen, so start there. White can play Qg6+, but Black ends the fun with NxQ. White has no other checks that go anywhere, so you turn to that knight and a way you might capture it: RxN. Once his knight has been taken, Black has to either accept its loss or recapture with QxR. Now reconsider Qg6+, drawing protection for the queen from the bishop on f7 and forcing Black to put his king on h8; and then Qxg7#, drawing protection from the bishop on c3 and mating (or Qxh6#, with the c3 bishop providing a pin). The initial capture RxN actually removed two guards: the knight, which was eliminated, and the queen, which was dragged away from the protection of the g7 square that White turned out to need.

Black should understand the consequences of recapturing with QxR, and if he does he will look for other options—and particularly for some way to provide a substitute guard against the mate threat, such as Bf5 to replace the knight’s protection of g6. This loses the bishop to QxB+, but then gives the king a chance to move to h8 and end the immediate threat of mate. (Mate follows sooner or later anyway, as Black now is badly overmatched in material and his king’s position is cramped.)