With Black's king surrounded, it looks as though it ought too be easy, but most mating attempts run into the same problem: as soon as White moves his knight from f6 to open the line from his queen to the black king, he runs into a discovered check on the f-file. White could eliminate the troublesome knight with 1.Rxf5, but that abandons the rook's defence of the d7-pawn and lets Black survive until beyond move three with 1...Rxf6.
What White needs is either another way to get at the black king with his queen, or a way to get his own king away from the problems on the f-file. The solution accomplishes this in an astonishing manner.
White begins 1.Ke1!! apparently walking into a variety of ambushes; but look what happens. The threat is 2.Ng4+ Nxh6 3.Ne5 mate. If 1...c1=Q+ 2.Qxc1 h1=Q+, White mates with 3.Bg1. (Or if Black checks with a rook on e8 on the second move od this line, then 3.Be3 is mate).
If Black tries promoting his pawns the other way round with 1...h1=Q+ 2.Qxh1 c1=Q+, then 3.Rd1 is mate (or 2...Re8+ 3.Re5 mate). Finally, and perhaps most difficult to spot, if 1...Rae8+ 2.Nxe8+ Rf6, White mates with 3.d8=N!
The idea of walking White's king into a barrage of checks is reminiscent of another classic three-move problem by Sam Loyd.
Here White is ready to deliver discovered checks on the fifth rank and on the long black diagonal, but his difficulty lies in providing a guard for the e4 and d4 squares. The main line of the solution is quite extraordinary: 1.Ke2!! f1=Q+ 2.Ke3!! when any check from the black queen, bishop or rook is met by a discovered mate. The other variations I leave you to work out on your own.