When reading the ending of The Eumenides, it's important to bear in mind that this isn't just the end of a single play: it's the end of an entire trilogy of plays, stretching back to Agamemnon, and continued in Libation Bearers.
Without thinking about the back-story of these two plays, it's easy to look at the end of The Eumenides and say, "Okay, dude murders his mom, the judge lets him off the hook, and his accusers become productive members of society. Big whoop." Of course, you won't find the card-carrying Lit nerds at Shmoop saying anything like that (the Furies are some impressively nasty ladies; it's a big deal when they come around at the end of the play), but we can understand why somebody might.
Clearly, we need to take a look at the back-story. The previous two plays have one basic message: revenge may be fun at first, but it will come back to bite you. With no courts of law in sight, there doesn't seem to be any way for the characters in these plays to come to a peaceful end—except for the peace of the grave. What makes the ending of The Eumenides so important is that it breaks this cycle.
Remember: the Furies aren't just any old villains—they are, themselves, personifications of revenge. When they become guardians of justice, it is a major triumph for civilization over barbarity. Without this happening, Aeschylus seems to be telling us, the Athens (and, by extension, the ancient Greek culture) we all know and love simply… wouldn't exist.