If no one else answers your question, and you're still curious, I strongly recommend the book The Rise of Christianity by Rodney Stark. I think it does a good job explaining a lot of the growth of the Christian Church. He gets into a lot of the different appeals, but one place that really stands out is its appeal for women. It gave women more rights and power than most Roman institutions. Converted women oft brough their husbands, or at least their children, into the Church. Constantine the Great's mother, for example, was definitely a Christian, and Eusebius, in his official biography of Constantine, argues that Constantine's father was a secret Christian (he was relatively tolerant to Christians in the areas under his control) but we have no outside evidence of this. Christians also provided the social services you read about in the Beautifudes (caring for the sick, visiting people in prison, feeding the hungry, etc) which made them appeal to the poor (as well as people for whom caring for the poor was important--I belive frequently rich women). Further, they formed a clear, strong community that appeals to many (you see this a lot even today) and their commitment to their beliefs was demonstrated publicly and repeatedly (being fed to lions and what not) which surely made at least some people think, "Well, if they're willing to die for it, they must have something figured out." Also, they grew a lot through natural growth (they did not commit abortion or infanticide, so their fertility rates were higher, and they also adopted orphans regularly--exposed/abandoned children were most often girls, which meant even more women, who then tended to bring in their husbands or at least their children).

Well to give you a very simple summarized answer:

A lot of early Christians originally were Jews, so they were not necessarily following Roman pagan religions like Zeus. Jesus was a Jew, and Christianity was a direct evolution from Judaism. The whole point of Judaism is that there is only one God and eventually God will send a messiah to the earth to help steer people in the right direction. To many, Jesus was that messiah and thats how the religion was born. So early on at least, early Christians didn't convert to Christianity they just formed a divide between groups who thought Jesus was the messiah and those who did not.

It's also interesting to note that at this time there were A LOT of people claiming to be the messiah. It just happens that for whatever reason, Jesus built up a large following (mostly after his death) and it became a religion.

It's also important to understand how slowly Christianity grew. Jesus was born around the year 0, and was crucified (like thousands of others) and nothing really came of it. It wasn't until much after that he drew this following. There definitely were a lot of Christian martyrs - people willing to die before denouncing Christ - which helped spread the message. Keep in mind this was also a time when Christians were being persecuted heavily for being heretics - It wasn't until 313 when Christianity stopped being illegal in Rome when Emperor Constantine Converted, and it wouldnt become the official religion of Rome until 380.