Where It All Goes Down
Temple of Apollo at Delphi, Hill of Ares (Areopagus) in Athens
The setting of The Eumenides is not only interesting in itself, but it also provides you with a counter-example to throw at any literature teacher who paints a narrow-minded and inflexible portrait of ancient drama.
Why would this even come up? We have Aristotle to thank for that—or, really, not Aristotle, but the way his work was misinterpreted starting in the Middle Ages, and (unfortunately) continuing to the present day. The idea that comes out of this misinterpretation is that of the "unities," sometimes called the "classical unities." These are (1) "unity of action," (2) "unity of time," and (3) "unity of place."
The first of these, which actually does come from Aristotle, states that a play should concern itself with one significant action. The second two, which are more of a later invention, state that a play should take place in one location and should not depict an action that last longer than one day.
Now, if you look at Agamemnon and Libation Bearers, (Parts I and II of the Oresteia trilogy), you'll see that they actually match up with this plan pretty well. Agamemnon centers around one action (the murder of Agamemnon and Cassandra by Clytemnestra), takes place in one location (outside of the palace in Argos), and lasts basically one day (beginning before dawn, with the messenger's speech). Then, Libation Bearers centers around one action (the murder of Clytemnestra and Aegisthus by Orestes), takes place in more or less one location (it shifts a bit from the tomb of Agamemnon to outside the palace in Argos), and takes place in one day.
Unlike the previous two plays in the trilogy, The Eumenides majorly mixes things up. You could sort of say it centers on one action—the trial of Orestes—but what about unity of place? In fact, the scene shifts, starting in Delphi and ending up in Athens. Also, because Delphi and Athens are more than a day's walking distance from each other, the time clearly lasts longer than one day as well. So yes, if anyone starts lecturing you about the "classical unities," you can definitely point to The Eumenides as the play that breaks the rules. It's the James Dean of Ancient Greek drama.
How does all this rule breaking relate to what The Eumenides is all about? Well, we talk about this play's main theme as breaking the cycle of violent revenge depicted in the previous two plays; changing up the setting could be one way of symbolizing that. Also, because we've been hearing so much about the oracle of Delphi throughout the story, it kind of makes sense to actually take us there and show us what it's all about.
As for things ending up in Athens, which not-so coincidentally acts as the enlightened city that clears up all that nasty blood-feud business, well, Aeschylus and his audience sure liked giving themselves pats on the back. This was a little hometown pride coming out—it's the Ancient Greek equivalent of Chicagoan Veronica Roth setting the Divergent trilogy in Chicago.