Because the Russian title, Бесы, means "demons," and the book takes as its epigraph the passage from the Gospel of Luke in which demons come out of a man and enter a herd of pigs. The Possessed is the same title by another route: it is the title of the early English versions and of the French Les Possédés, while more recent translations made from the Russian often use Demons or The Devils.
The meaning of the title appears at the end, in the mouth of a character in fever. Stepan Trofimovich, ill in an inn, asks Sofya Matveyevna, a seller of Gospels, for the passage about the pigs and applies it to Russia: the demons that come out of the sick man and enter the pigs are all the sores and all the filth that have piled up over the centuries "in our great and dear patient, in our Russia."
The common reading says the demons are ideologies, which enter men and drive them into the abyss. It is not false, but it stops short of a part the novel explains on its own: Stepan includes in the account his son, Petrusha, and himself, "perhaps the first," at the head. The allegory stops being an accusation and becomes a confession.
The Gospel ends with the healed man, sitting at the feet of Jesus, clothed and in his right mind. Dostoevsky gave the same image as the theme of the book in a letter to a friend: the demons have left the Russian man and entered the Nechaevs, and the healed man sits at the feet of Jesus.