1 b6+! Kb8! are the first two moves. White has at most a draw after 1 h4 axb5, and actually loses after 1 bxa6 b5!, and Black loses after 1 b6+ Kxb6 because the White queen on h8 will cover a1 in the ensuing race. Why is 1...Kb8! any better? Well, after 2 h4 a5 3 h5 a4 4 h6 a3 5 h7 a2, the fun begins. (This is actually where Joseph's original composition began.)
6 h8(Q) a1(Q)! reveals Black's idea, since 7 Qxa1 is stalemate. Note that if the queens are traded off, the king and pawn endgame is a draw. Heroic measures are needed: 7 Qg8! Qa2! 8 Qe8! Qa4! White cannot go to f8 because after ...Qa3, the Black queen has a check on d6 which cuts off the King's escape squares on the seventh rank. And he cannot go straight to e8 because 7...Qg7 draws.
The final act is 9 Qe5+ Ka8 10 Qh8! The stalemate defense is scotched and Black has no way to cover the King's escape routes on c7, d7, and e7. White wins.
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