Some of the stuff on the Earth’s surface is loose, mainly ocean water. The moon sits out there in space, going slowly around the Earth. The Moon’s gravity says “come to me, Earth oceans.” Friction between the firm Earth (the land, the seabed) and the loose Earth says “Nu uh, you’re coming with us, ocean.” So the ocean water is pulled in two different directions. Thus, the water’s gravity calls back to the moon, “Howzabout you come with me instead?” The moon’s all like “K.”
So the moon gets tugged along. But wait! At any given time, the Moon is being pulled along by this ocean bulge (the tide) in only one direction. It’s being pulled forward and down. Well, that little, extra “forward” velocity takes it further out into space before the rest of the Earth can rein it back in.
When you sum all these instantaneous nudges over a billion years, you get a higher (i.e. farther away) orbit.