There's a lot more that can be written about this. As a start, you might find interesting the FAQ section "The Rise of Christianity" , which has several interesting pointers. The ones that look most appropriate are three due to /u/talondearg that, in a mixture, deal with "how" and touch on "why":

• "How did Christianity go from a small, hated cult to the state religion of the Roman Empire? What made it so different than the hundreds of other small cults throughout the Empire?"
• "Why did Christianity, and not another contemporaneous cult, spread so rapidly within the Roman Empire?"
• "How significant was Constantine's conversion to Christianity?"

Slightly tangentally, there's also "Was there much resistance by Roman citizens to keep worshipping Roman deities in the post-Constantine Rome?" by [deleted].

(The others, including the AMA, do not seem to me to address your question.)

But this is not to discourage discussion. More questions, data, and debate are welcome.