When answering this question, we must understand first of all that it is not possible to know for sure what role Rome played in the rise of Christianity. We know that Christianity arose in Rome and so we look for aspects of Rome that seem conducive to that rise. But we cannot know if those aspects really mattered. All we can do is speculate in the most informed way possible.
I would speculate that there were two main aspects of the Roman empire that played a part in the rise of Christianity. The first of these aspects is the Roman tradition of religious tolerance. We know that Rome is famous for having persecuted Christians, but that was mainly after they became a bigger group. As the faith arose, it was protected to some degree by the fact that the Romans practiced all sorts of religions. This rise would have been less likely to occur in a society in which only one religion was tolerated.
The second aspect of Rome is its sheer size and connectedness. Christianity, of course, arose in what is now Israel. It then spread to Rome and Greece and places in what is now Turkey very early in its history. This spread was facilitated by the existence of the Roman Empire. Rome offered very secure connections between places like Jerusalem and the city of Rome. It made it possible for the idea of Christianity to spread and for networks of Christians to arise.
Thus, we can speculate that the size and connectedness of Rome, along with its relative religious tolerance, allowed Christianity to rise.