If the defending king goes to the long side, the rook is left with the short side and cannot keep checking distance. The attacker drives the king away along the back rank and shelters next to the pawn.
Compare this with the previous lesson. Here Black's king is on d8, the long side of the e-pawn, and the rook on a1 is on the same side as the king. The rook would like to check along the a-file, but the king on d8 is in the way of its own defence, and the files to the right of the pawn (f, g, h) are too few for a checking distance.