Sean W. answered 01/24/20
Specialized tutoring for US History and Civics
For the quick and dirty answer please scroll to the bottom, if you are interested in how it came to be here is a summary of the trilogy:
First, a little history as the identity of the Furies is important:
The Furies gain the name Eumenides during the third play of the Oresteia titled Eumenides. Prior to this they are known as Erinyes, or the "angry ones" and they are vengeance unleashed; older than the Pantheon of gods, and born of the sin of violence upon a father by his son: Kronos the Titan castrates his father Uranus, the blood from the wound falls to the earth and from that blood sprung the three Furies: Allecto (the unceasing or the unceasing anger), Megaera (the avenger), and Tisiphone (the jealous). Their job is to avenge those who are unable to right the scales, usually someone killed by a family member.
Long story short, in the first play Agamemnon and his brother Menelaus want to sail to Troy to rescue Helen, Menelaus' wife who was taken away by Paris. Agamemnon organized the princes of Sparta and along with 100 of his own ships gather at the port of Aulis in Boeotia to sail towards Troy. Unfortunately the winds are calm and they cannot set sail. Agamemnon and Menelaus are instructed to sacrifice Iphigenia,Agamemnon's daughter by Clytemnestra, in order to appease the goddess Artemis whom Agamemnon had somehow insulted. He dutifully complies, the winds turn favorable and they set sail. 10 years later they return.
While they had been away Clytemnestra engages in an affair with Agamemnon's rival, and cousin, Aegisthus. The two plot to kill Agamemnon and after enthusiastically welcoming him home and encouraging him to return to his palace where she and Aegisthus have set a trap for him: they had placed a tapestry across the ground far too fine for a mortal only fit for a god, his walking over this offends the gods due to hubris, or pride. While he is soaking in his bath Clytemnestra sneaks in and kills him, marries Aegisthus making him king of Mycenae. She defends herself before the Elders by saying that this was vengeance for the murder of her daughter as recompense for his angering the gods. The Elders are concerned because Orestes, they say, will come for revenge.
That was play one, Agamemnon. Play two, The Libation Bearers picks up a few years in the future. Apollo commands Orestes to even the scales. So, he and his companion disguise themselves as strangers and start towards the palace. On the way there they stop at the tomb of Agamemnon to make offerings when a group of women head towards them. They hide and hear Electra, Orestes' younger sister, curse Clytemnestra and plead for the swift return of her brother to take vengeance. They come out of hiding and reunite. Together they come up with a plan to kill their mother. They arrive at the palace and gain entrance claiming to have word on the fate of Orestes. Clytemnestra is the first to hear that Orestes is dead and sends Orestes' nursemaid to fetch Aegisthus. She tells him that he must meet with the strangers and he must do so alone. Instead of a message he is promptly killed by Orestes. Clytemnestra figures out who the strangers are and begs Orestes for her life, Phylades (Orestes' companion) reminds him of Apollo's orders and he slaughters her. He is then tormented in his dreams by the furies for the unforgivable sin of matricide. He then runs to Delphi and the protection of Apollo.
Now for the third play Eumenides. Orestes arrives at the temple at Delphi but the furies will not leave him alone. Apollo orders him to flee to Athens and the temple of Athena for protection, which he dutifully does. Unfortunately for him Clytemnestra's ghost tells the furies where he has run off to so they hound him even there. He pleads to Athena for help as he committed no sin as he was acting to right the wrong of his father's murder. Athena comes back from Troy, listens, and decides that this is too complicated a case for even a god to decide. So she comes up with the idea of creating a jury of twelve of Athens' finest men to vote either for the furies or for Orestes. Before things get started she informs the twelve that if it comes to a tie that she will vote for acquittal. The votes are tallied and they have a split jury, Athena casts her vote and Orestes leaves for Argos. The furies are livid and threaten to destroy Athens as a substitute.
Now here comes the direct answer to the question:
She placates the furies and renames them the Euminides (the kindly ones) and sets them as protectors of Athenian prosperity.