These are good questions, and as you can imagine, much debated in the literature, with explanations ranging from acts of God (the paradigmatic example being the Jew quoted in Acts of the Apostles arguing that Christianity didn't need to be suppressed because if it flourished, it must be favored by God, and it would fail if it was disfavored by him) to enabling effective society coordination (particularly attractive for Islam: the horsebacked nomads managed to coordinate under Mohammed rather than feud, and did as well as the Mongols, with conversion then following from personal advantage and to escape dhimmitude) to arguments that it's just random drift (Carrier points out that the best estimates of the sizes of early Christianity as tiny even centuries after Jesus then necessarily imply that the annual growth rate must have been far tinier than commonly assumed).