The standard believers' line is something like "it's all part of God's plan throughout history, given a massive leap forward at the Incarnation, and which will eventually be achieved etc etc"
Unlike BlueH, I don't entirely subscribe to the idea that this was just the luck of the draw. There had been a number of competing religions in the ancient world which it has been suggested might have become the dominant religion of the Roman Empire, Mithraism being the most popular contender, since the latter seemed to be exported wherever there were Roman soldiers. So why did Christianity usurp Mithraism? I suggest it was not all to do with Constantine.
I don't want to go into my own explanations for Christianity's success in depth here, but I'd like to suggest two reasons for starters: unlike Mithraism, Christianity originally allowed women important roles to play (and even when the misogynistic Church fathers held sway, it was still less male-oriented than Mithraism).
The second reason, I suggest, was that Christianity was essentially an end-time religion, with an urgent expectation of the imminent Last Judgement (according to the words of Jesus and St Paul themselves). This sense of an imminent Apocalypse has somehow renewed itself throughout history, and continually added a bit of fiery pepper to the Christian diet, despite the efforts of the traditional Church (pace Origen and Augustine) to allegorise away such literal readings of the NT.
Any non-believers prepared to suggest reasons for Christianity's survival rather than luck?