Because Aphrodite promised him the most beautiful woman in the world, Helen. According to the myth, at the wedding of Peleus and Thetis, the parents of Achilles, Strife threw among the goddesses a golden apple meant "for the fairest". Hera, Athena and Aphrodite disputed it, and Zeus sent Hermes to take them to Mount Ida, where the Trojan prince Paris lived as a shepherd, so that he would decide.
Each goddess offered a gift. Hera promised power and dominion, Athena victory in war and wisdom, Aphrodite the love of Helen, who was the wife of Menelaus, king of Sparta. Paris gave the apple to Aphrodite, went to Sparta as the guest of Menelaus and took Helen to Troy. That is where the war comes from.
The Iliad does not tell this scene. It takes it for granted, and mentions it only once, in the last book, to explain why Hera, Poseidon and Athena do not want Hector's body returned: they had hated Troy since the day when Alexander, which is the other name of Paris, "insulted the goddesses when they came to his sheepfold, and praised the one who gave him grievous lust". The poet names neither Aphrodite nor Helen, and calls the choice blindness, the Greek word ate, which designates the error that leads to ruin.
The Paris the poem shows confirms the motive. In Book III, saved by Aphrodite from the duel with Menelaus, he tells Helen that desire has never so covered his senses, "not even when, having snatched you from lovely Lacedaemon", he carried her off in the ships. The choice on Mount Ida and the man who flees the fight for the bedroom are the same person.