White is behind in material and needs to make something happen. Let's approach this one with forcing moves. Step 1: Experiment with checks, including a brazen gesture such as Rd7, which invites Black to play BxR. Step 2: Consider the board as it would look afterwards; ask what lines would have been opened or closed as a result and what new checks you might then have. Here Black’s bishop would have evacuated the sixth rank, permitting White to play the check, and fork, Qf6+ (taking protection from the pawn on e5). The fork doesn’t quite work because Black’s rooks are connected and guard one another. So persist and ask what move Black would make in reply to Qf6+. His king would be forced to e8. Then Qxh8+ works for White after all; it's made safe because the connection between Black’s rooks has been broken by his king. More importantly, it’s a skewer that wins Black’s other rook: Black has to move his king back to the seventh rank, and now White has QxRa8—and a won game. Notice how goading Black’s bishop onto d7 removed d7 as a flight square for Black’s king—and thus forced Black to respond to 2. Qf6 with Ke8.
So you have a winning idea if Black responds to 1. Rd7 with BxR. But what if he doesn’t? Consider whether he has anything better. His only alternative would be to move his king to f8 or e8. Think about what your next check would look like either way. If he plays Kf8, you have QxRa8—mate. If he instead tries Ke8, 2. QxRa8+ no longer works because Black has KxR. That’s okay, though; instead you play 2. QxBe6+, forcing Kf8; then 3. Qf7#. So Black is required to play 1. …BxR in the first place to avoid mate. (These trains of thought are worth reinforcing until they are clear.)
Stepping back and looking at the original position, observe the open diagonals leading to Black’s rooks. See how they invite the idea of a queen fork, especially with the king on a center file and especially with a friendly pawn in the center poised to protect your queen. The basic forking possibility (Qf6) is a little elusive at first because the rook on h8 doesn't become loose, and thus doesn't become a good target, until later. You might think like this: if I could get my queen onto f6, it would fork Black's king and rook, and the rook would become loose if the king were forced by the check to step back onto the eighth rank. So if only I could get my queen over to f6...)
The other lesson to take away is the value of considering bold moves like Rd7+. The move looks counterintuitive because it loses a rook on the spot and you already are behind in material. But this won’t stop you from experimenting with the move so long as you remember that the frequent purpose of such checks is just to force changes on the board that make forks or other tactics possible. That probably is the easiest way to see the solution here: not by spotting the forking idea from the outset, but by experimenting with checks you can give and then with new checks that become possible after your opponent's replies. This leads you to Rd7 as a first move and then Qf6—at which point the fork comes into view.