So what about 1. e4? It stops the check on e2 and gets another pawn rolling. The trouble is that Black replies 1 . . . Rg8 when nothing can stop Rxg7 with a draw.
White must find a way to counter the threat of Rg8, and the natural idea is to get the king off the g-file. After, for example, 1. Kf1 Rg8, White wins simply with 2. f7. Before going on, let's narrow down White's first move options. 1. Kf3 or Kh3 would be met by Re3+ with the usual kamikaze rook checks to follow. 1. Kf2 and Kh2 are met by Rxe2+, so that leaves only 1. Kf1 and Kh1. 1. Kh1 is obviously nonsense, so let's analyse the other move.
1. Kf1 threatens f7, so Black himself must threaten a check. Let's try 1 . . . Rd8. Now White has a repeating mechanism to make progress: 2. Kf2] Re8 3. e3] Rd8 4. Kf3 Re8 5. e4, gradually edging forwards, avoiding rook checks. Encouraged, we continue 5 . . . Rd8 6. Kf4 Re8 7. e5 Rd8 8. Kf5 Re8 9. e6 Rd8 and now White finishes it off with . . . well, that's the problem. He can't finish it off. 10. e7 or 10. f7 allow Rd5+, while 10. Kg6 Rg8] is a draw again with Rxg7+ unstoppable.
So it has to be the obviously nonsensical 1. Kh1]] when play continues just as before, 1 . . . Rd8 2. Kh1] Re8 3. e3] etc) with the king slinking up the h-file until 8. Kh5 Re8 9. e6 Rd8 10. Kh6]] Black can only wait with 10 . . . Re8 when 11. e7 Rc8 12. Kh7] Rg8 13. e8=Q+] Rxe8 14. f7] finishes him off. A superbly crafted composition.