Why do the three envoys visit Achilles to argue and how does Achilles respond?From The Iliad
The envoys are sent to Achilles in Book 9 of The Iliad on behalf of King Agamemnon, in the hope that they will be able to persuade Achilles to return and continue fighting. At this point in the story, the Achaean troops have been driven back to their ships by the Trojans, and Agamemnon fears all is lost, but Diomedes and Nestor urge Agamemnon to persevere, saying that the war can still be won if they can persuade Achilles to rejoin them. As such, an embassy, including Odysseus, Great Ajax, and Phoenix, is sent to Achilles with Agamemnon's proposal, offering him a huge number of gifts as a condition of his return.
Achilles, however, rejects the offer outright when Odysseus presents it to him. He says that he intends to return to his homeland and live a long life there, which would be preferable to dying in a burst of glory with the Achaeans. He even asks Phoenix to join him, but Phoenix earnestly implores him to stay, reminding Achilles of the story of Meleager, which is an illustration of a warrior responding favourably to pleas from his friends. This tactic is unsuccessful, however, and the embassy returns to Agamemnon without reaching an agreement with Achilles.
The episode you are talking about occurs in Book 9 of The Iliad. You can look there for more information.
Basically, the three envoys are coming to Achilles to ask him to help the Achaians, who are losing to the Trojans. Achilles has been angry because Agamemnon took Briseis away from him. Because of this, he has been unwilling to help.
The three envoys promise to give Briseis back along with gifts and an apology. Achilles refuses and says that he will not fight.
As it turns out, he won't join the fight until Patroklos is killed.
The three envoys come to Achilles in Book 9 of the Iliad.
At this point in the epic Achilles, greatest of the Greek warriors, has withdrawn from battle after having Briseis, a slave woman he had claimed as a prize, taken from him by the Greek King Agamemnon. After his withdrawal, the Greeks start to lose the war, forced from Troy all the way back to their own ships. The Trojan army is threatening to attack again and burn their ships, stranding them in hostile country.
After advising with the other Greek leaders, Agamemnon seeks to remedy his offense, offering to return Briseis to Achilles along with other great gifts, including the hand of his daughter in marriage.
Agamemnon sends three envoys with his offer: Odysseus, cleverest of the Greeks; Phoenix, a wise elder and former tutor of Achilles; and Ajax, greatest of Greek warriors besides Achilles himself.
Each envoy presents a different argument. Odysseus begins, detailing the gifts Agamemnon offers and calling on Achilles to accept, both because the gifts more than make up for what was taken, and out of pity for the battle-weary Greeks. He ends with praise for Achilles's skill, asserting that he could certainly drive the Trojans back and personally defeat the Trojan champion Hector.
Phoenix speaks next. He reminds Achilles of their history, which begins in Achilles's earliest childhood when Phoenix was a guest of his father. He recounts a touching story that a young Achilles would not eat or drink except in Phoenix's lap, and would spatter Phoenix's tunic with food. He calls on Achilles to return to the war and defend his people. He ends with a story of a hero of his own generation, Meleager. He too won great victories and was poorly treated, like Achilles. When his people were attacked he also refused to fight, relenting only at the last minute, finding friends slain and lands destroyed. He calls on Achilles not to make the same mistake.
Ajax is the last to speak. His speech is shortest of the three, warrior to warrior and man to man. He reminds Achilles that the conflict began over a single enslaved girl, and among his gifts Agamemnon has offered seven. He tells Achilles that, as was Greek law at the time, worse offenses, even the murder of a family member, could be paid off with a blood-price far less than what Agamemnon had offered. Finally, he recalls to Achilles that he was the best-liked man among the Greeks, and that in failing to fight he is betraying that friendship.
Achilles refuses each in turn, but each refusal is slightly less than the last. He contradicts Odysseus completely, condemning Agamemnon's offer as worthless because Agamemnon is unworthy of trust. He declares his intention to gather his ships and depart, hoping that his ill treatment will serve as a warning to other Greeks not to put their faith in Agamemnon. In refusing Phoenix, however, he is less certain, calling on the old man to put aside his loyalty to Agamemnon and remain with him, spending the night in his own tent and discussing in the morning whether to stay or go. Ironically, it is Ajax, who says least, who arguably has the greatest success. Achilles's response to him is that he and his ships will remain in Troy for the present, but that he will not fight unless Hector and his Trojans reach his personal ships.
Phoenix accepts Achilles's offer and remains. Ajax and Odysseus return to the Greek leaders and explain that Achilles has rejected Agamemnon's offer and will not fight. The fact that Achilles stands by his refusal is vital to the Iliad. The Iliad chronicles the idea of honor and its consequences. Achilles's offense comes less from the fact that Agamemnon had taken someone dear to him than because he felt Agamemnon had demeaned him, demonstrating his superior authority by taking something Achilles valued. In doing so, Agamemnon dishonored Achilles.
He refuses to fight unless the Greeks attack his ships.