With the white knight covering c4 and c6, and the rook on d7 guarding d5 and d6, all White needs is a check from his other rook and Black is mated.
Rb6, Rhd6, Re6, Rf6 and Rg6 threaten, respectively, Rxb5, Rd5, Re5, Rf5 and Rxg5 mate; we need to find Black's defences to each of these to piece together the solution.
After 1.Rb6, Black has 1...b1=Q, defending b5. So how about 1.Rhd6 (threatening Rd5 mate) and after 1...Bb3 (protecting d5) the b-file is blocked and we can play 2.Rb6!
That would be fine, except Black plays not 1...Bb3 but 1...Bf3! to defend d5.
So what about 1.Re6? If Black then plays 1...Nf3 to stop Re5 mate, we continue 2.Rd6! when Bf3 is no longer possible, so he has to play 2...Bb3, after which 3.Rb6 forces mate.
Yes, but after 1.Re6, Black plays 1...f3! opening the bishop's line to protect e5.
Well, we could interfere with that by starting with 1.Rf6, and when he plays 1...Ng3 to prevent Rf5, we reply 2.Re6 and 2...f3 no longer works.
Yes, but what if he defends with 1...Bg4! after 1.Re6? All the required defences then work again. But now we're almost there. It has to be 1.Rg6!! Rg4 (defending g5) 2.Rf6! Ng3 (Bg4 is no longer possible) 3.Re6! Nf3 (f3 doesn't work) 4.Red6! Bb3 (Bf3 is illegal) 5.Rb6! and, with b1=Q ineffective, Black can only choose between 5...Rb8 6.Rc6 mate and 5...Bc4 6.Nb7 mate. A beautiful concoction.