Once the rook pawn reaches the seventh with the king beside it, cross-checks and queen trades become available and the attacker wins. The extra rank makes all the difference.
The rook pawn becomes dangerous only when it reaches a7 with the king on b6, b7 or b8 beside it. Black's queen on c1 must prevent a8 by controlling the long diagonal or the eighth rank, and that fixed duty gives White the time for queen manoeuvres with check.