Aegeus is the King of Athens. He passes through Corinth hoping that the Oracle of Delphi will help cure his inability to produce children. Medea discovers his dilemma and agrees to give him some remedial herbs if he promises to grant her asylum in Athens. Aegeus agrees, especially since the two of them are already old friends as it is. He is unaware, at this point, of how much Medea has changed since he last knew her, when she was an unmarried princess.
Aegeus's sterility is a major issue, since he is a king. Producing children is the only way he can continue his direct line. As the previous answer mentions, this highlights the cruelty of Medea's murdering her children with Jason: she is denying him heirs and destroying the physical embodiment of the love they once shared.
On a plot level, Aegeus is Medea's way out of paying for her crimes, at least in the sense of paying the law. She will be protected in Athens, because Aegeus cannot go back on his vow.