What happens to dwellers in training once their stat is maxed out?

They remain in the room until you move them somewhere else. They stop training, however they're not quite doing nothing. They still count towards the number of dwellers assigned to the room. This increases the training speed of any other dwellers in the room who haven't also maxed that stat.

Also note that dwellers only increase their SPECIAL stats when they finished training and you then touch the stat icon above their head. So you'll be interacting with the dweller anyways once they have maxed their stat.

They stay in the training room taking up space, but doing nothing. Instead of a time underneath them, it will say Max Level Reached.