Why did Christianity take hold in the ancient world?

This is a big question which involves many factors.Many historians have wrestled with this question.
It first appealed to slaves and women. Some of these women were wealthy, and supported local communities. In some of the Gnostic sects women were allowed to preach, though that was eliminated by about the year 280, when the founding fathers argued successfully for a unified doctrine and creed.
Christians were persecuted because they refused to 'worship' the Emperor, which made them atheists (the rite of communion was rumored to be cannibalistic). 'Worship' in this case meant the formal performance of a simple rite, which involved going to a temple, burning some incense, giving a gift (it could be as simple as a bouquet of flowers), and praying for the Emperor's health. That Christians saw this as an act of sacrilege and idolatry says a great deal about how fanatical and simple minded they were. Some magistrates begged them to reconsider, but most refused, even if it meant sacrificing their own children. It's true that Christians were persecuted, but it's also true that they avidly sought persecution (as a sure path to Heaven).
Over time their fortitude in these public persecutions generated substantial sympathy for them, though the persecution of the emperor Diocletian almost wiped them out.
Christian propaganda was better than what pagan religion had to offer. Christians held out the hope of immortality and the reward of heaven, neither of which was available to ordinary people in pagan religion. Charity existed in the ancient world, but the Christians were better at dispensing charity and better at caring for orphans. The message of universal love and peace also appealed to many people.
There was a trend towards monotheism among the intellectuals of the day, and Christianity was fanatical about monotheism, which eliminated all the petty disputes and arguments of the gods and goddesses. It simplified things and seemed to make more sense to people.