The Adoption of Christianity by the Romans
It is true that Christians were hated by Romans before reign of Constantine. The circumstances changed when Emperor Constantine came to power in 306 AD. Constantine had followed Christianity for political reasons. One example is that he had declared that the victory at the Battle of the Milvian Bridge, 312 AD was due to Christianity.
Romans accepted Christianity as a religion because their Emperor accepted and patronized it. The acceptance of the Christianity by the emperor had not ended the pagan worship.
Christianity made no social distinctions. Everybody was free to accept it. Common people found it more pleasing than the pagan worship.
Christianity offered much more than the other Roman pagan religions. It offered a hope of better life after death. Inspite of the similar offer by the pagan religions, the appeal of Christianity was more attractive.
The Christian doctrines were understood by everyone as it was written in Greek or Latin. The increasing followers of Christianity might have influenced the Romans to accept it.