How does Aegeus' scene relate to the overall theme of Medea?

The Aegeus scene is important to the theme of the play in several ways. On the first level, it is central to the plot: it givies Medea a place of refuge which facilitates her escape from Corinth.

Next, it shows an important element of Medea's character. She is intelligent and skilled in arts that exist on the border between medicine and sorcery. Had she been a man, she would have been a successful professional. However, she is a woman. That means that despite her intelligence, knowledge, skills, and brave and independent character, she is ultimately dependent on men, including her father, Jason, Creon, to a degree, and now Aegeus. Part of the anger and desire for revenge at the core of her character is a reaction to this sense of dependence, especially after her betrayal by Jason.

It is also worth noting that this scene emphasizes the plight of migrants. In the Athens of Euripides' own period, citizenship was restricted to those with Athenian grandparents. Immigrants could only be "metics" (a sort of resident alien). They lacked the rights of citizens and—unless they were very wealthy— they were forced to live a somewhat marginal existence despite often being skilled craftspeople. Thus, this scene reflects the precarious position of the immigrant and foreigner in the face of xenophobia and how such migrants live uncertain lives, dependent on the whims of rulers.

What is Aegeus's role in Medea?

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.

What does Aegeus seek 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.