What does Aegeus want from Medea?
I have edited your question, assuming that you are asking what it is that Aegeus wants from Medea, not what he wants from himself. I will also tell you what Medea wants from Aegeus, because ultimately, they come to a mutually beneficial agreement!
When Aegeus arrives in Corinth and runs into Medea, he seems happy to see her and is very sympathetic to her situation. He agrees with her that Jason has greatly wronged her and their marriage vows by leaving her and marrying the princess of Corinth in order to raise his own status. The audience knows that Medea feels alone, is extremely angry, and that has formulated a plan against Jason (though we don't know the details yet). To make matters worse King Creon has given her a one day reprieve to find somewhere else to live, but he is banishing her from Corinth because he fears her magic and her anger towards his daughter and Jason. In the course of her conversation with Aegeus, she gets him to agree to provide her a safe haven in Athens. His only stipulation is that she must get to Athens on her own powers; he will not be connected to anything she does in Corinth or provide her aid in escaping. In exchange for this safe-haven, Medea tells him that she, through her magic, will be able to ensure that Aegeus and his wife will have a child, which up to now has not happened. Aegeus is anxious to have an heir, and willing to make this bargain with Medea to achieve his ends. Both people will benefit from this alliance and get what each desperately needs.