When the enemy rook alternates between checking your king and attacking your pawn, walk the king down the board one rank at a time; eventually the enemy king is driven to a square where the pawn-defending rook can be exchanged or the pawn captured.
White's c-pawn is on c7, one step from promotion, but Black's rook on c2 attacks it and the black pawn on h2 is about to promote itself. Simply promoting fails: 1.c8=Q? Rxc8 and the white king on a7 cannot recapture. Nor can White take the h-pawn, since 1.Rxh2? Rxh2 leaves the c-pawn lost too.