Aegeus offers Medea refuge in Athens, but he makes it clear that she must get there on her own. He is willing to help her, but he doesn't want to be actively involved in her actions. Aegeus is protecting his own position and relationship with Corinth, and he is doing this favor in return for Medea's magical help in his need for an heir. Medea, as shrewd part of the deal makes him promise that he will never cast her out to the hands of her enemies. She has plans for revenge, and knows that things are going to be awful by the time she needs to leave Corinth.
He doesn't refuse in the end -- we never hear anything more about him. We are to assume that the magical chariot with the dragons is Medea's means of transportation to Athens in the aftermath of the murders of the princess, Creon, and the children.
The actual myth has Medea come to Athens and continue to live her life.