Why does Achilles refuse to accept Agamemnon’s apology?

Nestor advises Agamemnon to make peace with Achilles so that he will rejoin the fighting. Agamemnon agrees that it was madness that made him insult their greatest warrior, and prepares an offer for Achilles. He will give the great warrior fabulous riches, including one of Agamemnon's own daughters as wife and seven of Agamemnon's citadels, if only he will return and "yield place to me, inasmuch as I am the kinglier" (9. 160). Nestor proposes sending Phoenix, Great Ajax, and Odysseus, as well as the heralds Odius and Eurybates.

The ambassadorial party goes to the Myrmidon encampment, and they find Achilles playing his lyre and sitting with his beloved companion Patroclus. The two men rise on seeing the party, and Achilles treats his guests with great courtesy, asking Patroclus to ready food and drink for them. A good meal is prepared, with sacrifices to the gods, and Odysseus makes his proposal to Achilles. He tells him that the Achaeans are in trouble and need their greatest fighter, and he gives Achilles Agamemnon's offer. The offer is repeated verbatim from Agamemnon's own speech until the end, where Odysseus leaves out Agamemnon's statement about Achilles needing to yield to Agamemnon's kingly majesty. Odysseus also adds one final, important argument: if Achilles still hates Agamemnon, he should rejoin the fighting out of pity for his friends and fellow soldiers, who are being slaughtered for want of their greatest warrior.

Achilles responds that he will not return, nor would he even if he were offered treasures far richer and greater than those offered by Agamemnon. The possessions, Achilles argues, are not worth his life. His mother has told him that he can either stay and fight and gain great glory, or he can return home and have a long life.