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.