With White’s king stuck and the d-file open, Black naturally would like to mate with RxR. The move is prevented by White’s queen on g4, so make that queen your focus: look for anything else it protects or any way you can harass it, recognizing that its range of motion is limited by its defensive duties. You also want to be aware of any loose pieces on the board, such as White’s rook on c2 here; such pieces make good targets for double attacks, and a largely immobilized queen makes for another. Black therefore plays 1. Qe4, forking queen and rook. If White plays QxQ, Black mates with RxR (White just has a useless interposition).
Are we done? Not quite, because you have to ask whether White has an answer: a place he can move his queen where it has protection and can protect both the mating square d1 and the forked rook on c2. As it happens, such a square exists in e2. But if White plays Qe2 Black has 2. ...QxQ, and after White’s recapture RxQ Black plays 3. RxR+ and mates next move (again White has a useless interposition). So White’s best reply to Qe4 is to move his queen to f3 where it continues to guard against mate, and thus to forfeit the rook on c2 to QxR. (If Black instead forces an exchange of queens, White recaptures with his g-pawn and now his king has a flight square.)