A demanding position. Inspect for unguarded White pieces and you find the bishop (and the queen, but focus on the bishop). Black has no safe way to check the White king and attack the loose bishop at the same time. Does Black have the makings of any mate threats—any pieces already attacking squares next to the king? Yes: the bishop at d7 attacks h3. Next: can Black’s queen attack h3 and the loose bishop at the same time? Yes, with Qh5.
That's the idea. But now Black plays through this sequence mentally and finds a problem. If White replies by moving his bishop (say, to b7 or f3), Black does not quite have mate with Qh3+. Here, unlike in the previous problems, the king has a flight square at g1 or h1, and then there is no way for Black’s queen to reach it while maintaining protection from his own bishop.
Still, the basic idea looks promising, so keep at it. Where a combination doesn’t quite work, experiment with move order and, above all, be thorough in examining every check and mate threat. We have two sequences to think about, depending on what White does with his bishop....
(a) Suppose first that White replies to Black's Qh5 by moving his bishop back to b7. We had been imagining that Black now would play Qh3 (and we found that it didn't work). But what other checks are possible for Black instead of that one? There is Qd5 (no good—loses the queen to White's bishop), and there is Bh3. In response to Bh3 White would have to play Kg1 or Kh1. You now ask what comes next, and answer the question by examining every check. There would be just one: Qd1―which leads to victory! How? After White plays RxQ, Black plays c2xR, promoting the pawn to a queen and delivering checkmate.
(b) Now let's imagine that after Black starts with Qh5, White moves his bishop to f3 (rather than b7). The initial idea for Black is the same: give check with Bh3. And again White is forced to move his king to the back rank. But this time the follow-up for Black is a little different. He can't play his queen to d1, at least not in one move, because White's bishop on f3 is in the way. So instead Black does it in two moves. He starts with QxB. This gives him a bishop on h3 and a queen on f3, with the obvious threat of mate via Qg2 next move. White's only way to prolong matters is Qb7, so that his queen protects g2. But now Black uses the same little sequence we saw in variation (a) above. He plays Qd1+; and when White replies RxQ, Black recaptures with his pawn, promotes it to a queen, and mates.
Of course the point of seeing all this isn't that any of it is likely to happen. You simply are establishing that if White tries to save his bishop after Black starts with Qh5, White ends up mated. So White's best play after Qh5—and the thing you should expect him to do—is to abandon the bishop (e.g., with f2-f3, which at least prevents Black's forthcoming capture from also giving check).
This is a harder problem than the previous ones, but it can be cracked with persistence in examining every check and mate threat, and then every check that would be possible in the subsequent position. And of course you must be alert to the possibility of promoting the pawn on c2. A pawn on the penultimate rank (or indeed any passed pawn) always is a mighty tactical factor; it requires close attention after every adjustment to the board.