How does pride lead to destruction in Homer's The Illiad?

Homer's Iliad is the story of the final weeks of the Trojan War, when the tide of battle turns decisively against the Trojans after a ten-year stalemate. The catalyst for the events of the poem is the headlong clash between the pride of Agamemnon, the high commander of the Achaean forces, and the pride of Achilles, the Achaean's pre-eminent warrior.
A plague sent by Apollo is ravaging the Achaean troops. Achilles suggests they determine why Apollo is angry, and it transpires that the plague was called down on the Achaeans by a priest of Apollo, grieved that his daughter, Chryseis, had been taken from him as a trophy. Agamemnon is the one who "won" Chryseis, and he is responsible for returning her to her father in order to end the plague. Angry that he must give up one of his "spoils" of war, Agamemnon insists that it is only fair that Achilles also give up his woman, Briseis, whom Achilles won in the same battle. This is not necessary, but Agamemnon is lashing out because Achilles's inquiry has made him, Agamemnon, personally responsible for the plague.
Achilles is outraged that Agamemnon is penalizing him, and refuses to comply. Agamemnon then asserts his authority as high commander and sends men to Achilles's ships to forcibly remove Briseis from him. Achilles, furious, retaliates by withdrawing from the battlefield, depriving the Achaeans of their best warrior. Other men try to mediate between Agamemnon and Achilles, asking the one to apologize for his grandstanding and the other to forgive and rejoin the army, but neither man will budge.
Achilles is not merely a warrior, he's effectively a weapon, and his absence from the battlefield seriously weakens the Achaean forces. Hector, prince of Troy, immediately senses the opportunity presented by Achilles's absence, and the Trojans launch an extremely aggressive effort to push the Achaeans back to the shoreline and destroy their army. In the weeks in which Achilles is not fighting for the Achaeans, many, many people die, the Trojans succeed in throwing the Achaeans right back to their ships, and then set the ships on fire to prevent the Achaeans from fleeing. The battle is at fever-pitch, and the Achaeans are almost certainly doomed before Achilles rejoins the fray.
The clash of Agamemnon's and Achilles's egos at the outset of the poem nearly results in the destruction of the Achaean army, and it's only when their egos are set aside in service to the cause of defeating the Trojans that the Achaean forces rebound and are able to defeat the Trojans.