Christianity had many competitors in its early days: the old Hellenic religion, Judaism, and the mystery cults of Sol Invictus and Mithras. There was also the Roman state religion, i.e. deification of the augusti, which one could practice in accordance with many established religions (with Christianity being a notable exception).

First, the spread of the teachings of Christ through the written word between learned people of clerical and priestly classes. Koine Greek being the intellectual lingua franca of a vast area (which was also highly urbanized and sophisticated by the standards of the time) allowed these writings to be spread very efficiently.

Christianity also had broad appeal, especially to the lower strata of society: slaves, laborers, the poor, etc. They may have been socially disadvantaged, but this group was numerically superior by a vast margin. Its central tenet and promise was that if you accepted Jesus Christ as your lord and lived by his teachings—which, for someone who is poor, humble and generally a good person, is not far out of reach—then you will be rewarded in the afterlife with eternal bliss. Life was tough for a lot of people, and the idea that they would gain entry to paradise if they persevered and kept their faith was highly attractive. Despite being God, Jesus lived as they did; he worked, he ate, he drank, he experienced pain and suffering, and he did it all for them; that's a powerful and deeply personal message when most gods you know live on top of a magic mountain and throw thunderbolts around for fun.

Perhaps most importantly, Christianity was a prosleytizing religion, unlike Judaism. The conversion of non-believers was a core tenet of the faith. Throughout the religion's entire history you have evangelists, missionaries and other holy men spreading the faith, many times risking their lives (and often losing them) to do so. That kind of hardcore belief and obstinance has a snowball effect.

By the late 4th century, when the Roman Empire adopted Christianity as its state religion, the church had imperial resources at its disposal; it spread, co-opting local traditions, gaining adherents and building upon its successes. Its deep association with the Roman state continued to thrive in the Eastern Empire long after the fall of the West. And while the Western Roman Empire might have failed, the church carried on many of its clerical, bureaucratic and administrative traditions in the new kingdoms that formed from its ashes.