White is down exchange and some pawns, and the King has no cover at all except a black pawn, which can move forward at any time. So there is an urgent need to find mate.
Black King is not safe either. It has three squares to move. So naturally Rc1+, taking away one square. Kb8. Now Queen check on b-file will force the King into a8 corner, but where to check, b5 or b4?
After King goes to the corner, we have two lines for checking the King, 8th rank and h1-a8 diagonal. But other than c6 is free, all other squares, c8, d5, e4, f3 and b7 are guarded by black Queen and Rooks.
If there are defenders, we should think about removing defenders. How do we distract, or lure away defenders? in what order?
First Qb5+ can be followed by Qc6+. That's it. But we can at least have perpetual checks. If we are in time trouble, and can't think through other plans, this will be our fall-back plan. Qb5+ is also very natural because it creates a pin on Rook and Queen. This pin is very weak and might backfire. If after Ka8, Bxf7, wishing to lure the Queen away, black can do b2+, discovery check; after white King is exposed, Rook on d7 will make another check to discover white Queen. Too bad! So if we can't find a mate, we need to explore another route.
After Qb4+, Ka8, we are actually seeing three squares that might lead to a mate, f3, e4 and c8. Black Queen is guarding c8 and e4, and Rook on e3 is guarding e4 and f3. Now you should see our plan, we will give up our Bishop first, then our Queen, and Rc8#.