As a Christian theologian (in training), my answer is that the expansion of Christianity was guided by the Holy Spirit. But that probably wouldn't be a proper answer for the subreddit.

However, there are many little things that helped Christianity. The first is the inherent urge to evangelize. It is at the core of the religion and expressed clearly in the religious texts (like Matt. 28:18-20). Other religions in the Roman Empire didn't usually have this same urge.

Second, as is noted elsewhere on this subreddit, the Christian religion was not ethnically limited. While this turned some people off, it also allowed the religion to not be limited in who they would accept among the ranks.

Third, perhaps most importantly, was Christianity's mercy within a hostile culture. Children were often left to die of exposure, especially women. Christians would often come and rescue these babies, raising them as their own. Contrary to what people believe today, Christianity was also comparatively very egalitarian both among the classes and among the sexes. The early church was largely composed of slaves, lower class, and women. This appeal to a massive audience within the Roman Empire helped the spread.