If you check Black’s available moves, you will find that all but two of them allow mate in one. So you need to find the two defenses, and work out how White can set up a mating reply for them.

White cannot mate immediately because of the pieces on the back rank, for example Rh4 mate is prevented by the bishop on d8. Likewise, the other pieces stop e3 mate, Qf5 mate and Nd5 mate. But the fact that each Black piece has a responsibility proves to be Black’s undoing, because they can hardly move without stepping on each other’s toes. After 1…Re7, for example, White can play 2 Rh4 mate, while 1…Re6 blocks the other bishop and allows 2 Nd5 mate. Indeed, in the diagram position Black has only two moves which cannot be met by immediate mate, namely 1…Bg5 and 1…Re3. If only White can find a way to prepare mates for these moves, then Black will be obliged to fatally weaken himself. What weakness is created by 1…Re3? The answer can only be that it blocks e3, and therefore the bishop on g1 is freed to deliver mate by Bh2. Thus the first move must be with the rook on h2, but should it move to h1 or h3? The answer is provided by the other possible defense, 1…Bg5. This blocks g5 and frees the queen to mate by Qh2. So the solution is the unlikely-looking 1 Rh1! There is no threat, but nevertheless Black has no defense. Here is the full list of variations.
1…Re7/Rf6/Bxc7/Bh4 2 Rh4 (or Rxh4)
1…Re6/Rf7/Bxh7/Bd5 2 Nd5 (or Nxd5)
1…Re5 2 Qg4
1…Re4 2 fxe4
1…Re3 2 Bh2
1…Rxe2+ 2 Nxe2
1…Be7/Be6 2 e3
1…Bf6/Bf7/Rf5 2 Qf5 (or Qxf5)
1…Bg5 2 Qh2
1…c3 2 Nd3