Your eye should start at Black’s back rank since you are close to mating with RxR; only the queen far away on a3 prevents this. So turn to the queen and you ask how you can take advantage of its limited mobility—whether you can attack it, and whether it protects anything else you might attack. You see the rook on b2 that the queen ostensibly protects and so think about ways of going after it. 1. Qc1 isn’t the answer because Black has too many good replies. He can defend the rook with his c-pawn, or better still can simply play RxR on his back rank.
But notice that 1. Qc3 is an entirely different matter. Now White’s queen attacks Black’s queen as well as his rook; if Black plays RxR on his back rank, White has QxQ. Yet if Black instead plays QxQ he is mated right away with RxR. Black does have the option of Qc5, where the queen moves to safety and still protects f8—but now White wins a rook with 2. RxR+, QxR; 3. QxR. (Presumably you see why starting with 2. QxR is no good.) The back rank mate never does come to fruition, but the pressure created on Black by the threat of it causes material losses elsewhere. We see again how even a mate threat that doesn’t quite work can pay dividends.