How did Christianity emerge and then spread to become the official religion of the Roman Empire?
There are a confluence of reasons for the growth of Christianity in the Roman Empire. However, let me mention three main ones.
First, Christianity, unlike other religions in the ancient world, was a missionary movement. They actively sought to convert people. If you think of the work of Paul, it was really a missionary endeavor. Moreover, all this activity took place in the Roman world.
Second, Christianity was persevering. Even in the face of persecution, Christians did not give up or give in. Moreover, they had great apologists that made reasoned arguments for their faith, which persuaded many. In connection with this, a number of people in the Roman empire found in Christianity new identity and status - women and Roman veterans. Wayne Meeks, the Yale scholar, has convincingly shown this point.
Finally, Constantine defeated Lactantius at the battle of the Milvian Bridge. Before this battle, Constantine is said to have seen the sign of the Chi Rho in the sky with these words: "In hoc signo, vinces." He attributed this to Christ. The Chi Rho are the first two letters of Christ in Greek. When he won, he converted to Christianity, and on account of this many others converted to Christianity. This was the definitive step. After this, there was no turning back - not even Julian could bring back paganism.