Black has two checks to examine: RxR, which then loses the rook and amounts to an unhelpful exchange; and Qh3. White’s only legal move in reply to Qh3 is Qh2—an interposition. It’s hard to see how Black could take advantage of it, since the pinned queen is attacked once and protected once; there is no way to bring Black’s rook into the fray effectively. Don’t give up, though; the White queen’s move to h2 has greatly limited the king’s liberty, so think about another check: Qf3. How would White respond? The king can't move and Black’s queen can’t be taken, so White would have to interpose either his queen (with Qg2, losing the queen to RxQ) or his rook with Rg2. Now the rook would be pinned. This time would Black have a way to throw another piece at the pinned target? Yes; with h4-h3. The rook gets taken by the pawn on the next move. (If White plays Qxh3, Black replies QxQ.)
The moral of this position is that if a first check causes changes but creates no tactical openings, see what new checks are available and assess the position that results as if it were on the board in front of you. Repeat as necessary.