Originally posted by caffienexThat leads to at most a draw, pieces exchanged and white is out with his king. With h4, black prevents the king from going to g2 where the rook would no longer be pinned, unable to move which prevents the other rook from attacking or defending elsewhere, so its effectively pinned. So h4, white tries g3, say, to try to get the king up to g2 but then black goes h3 with pawn, preventing the king from going to g2 so then its just pawn pawn pawn, etc till there are no more pawn moves then white has only losing moves left, either move the bottom rook or move the king, either way a rook is lost and the game.
Bishop takes Rook wins.
Besides the already given solution 1.h4 g3 2.h3 and letting white run out of moves, it may be useful to indicate that after h4, h3? (which tries to block black from h3) doesn't work for white either because he cannot free himself without playing the g-pawn, this time with immediate loss of the rook after hxg3 or hxg3 e.p.