To move in is to bring your boxes and start living in a new home. The day the keys change hands.
It is used without an object, and the matching phrase is move out. Moving in with somebody is a distinct milestone in a relationship and has its own weight in conversation. There is a second sense meaning to approach threateningly, since police move in on a suspect and a competitor moves in on a market. A third phrase, move in certain circles, means to mix socially with a particular set.