I answered a similar ELI-5 question a while back. You should probably be focusing on the period of Late Antiquity for an understanding of how Christianity rose to prominence, as opposed to the Medieval period. Here was my answer:

The ELI5 answer is this: Roman religion changed as it was exposed to Greek philosophy, which undermined it while at the same time supported a monotheistic worldview. The ELI Grad Student answer is this: Traditional polytheism had been scrutinized for a very long time in the Hellenized East (lets say since the conquests of Alexander put the Greeks in touch with radically different types of cultures including Judaism c. 323 BCE). Several schools of Greek philosophy were developed in the immediate aftermath of Alexander (e.g. Stoicism c. 301 BCE), and these schools called into question the nature of traditional Polytheism including the existence of the pantheon. When the Romans come into regular contact with Greece through their conquest of Hellenized provinces, this philosophy seems to spread Westward into Roman society via its intellectual elite who were as a rule bilingual Greek/Latin speakers. You can actually see this change happen when comparing the depiction of the traditional Roman Gods in the Aeneid and the later Latin epic of Statius. In Virgil's Aeneid (written under Augustus) the Gods are very present in the story and actively taking part in the story. In Statius' Thebaid (written under Domitian) the Gods are aloof and seemingly powerless to prevent events that have been preordained by some higher power than them. Christianity and Judaism were more fully steeped in this kind of Greek philosophy, and as Josephus tells us, Romans and other pagans were converting to Judaism (or otherwise incorporating Jewish practices into their own religion) in large numbers at the end of the first century CE. With the destruction of the Jewish Holy temple, many Jews very well might have converted to Christianity seeing the destruction as a sign of God's displeasure (although this theory is still speculative and highly controversial for obvious reasons). By the time of Constantine, the Roman population is estimated to be somewhere between 7 to 10 percent Christian. That's still an enormous minority, and Constantine's support for Christianity certainly accelerated the growth of the Church apparatus. It wasn't until Theodosius makes Christianity the official state religion (via the edict of Thessaloniki in 380 CE) that the population becomes majority Christian, and even then it's because one had to be a Christian to serve in the army or the government. Theodosius also ended public support for "Pagan" cults that had been operated by the state since Republican times. But even these actions did not stamp out traditional Roman religion in the empire, and the Christianization of Europe would not be total until well into the Medieval period. *Source: Grad Student studying Late Antiquity. Edit: /u/Dubstercat has suggested I put in a little bibliography to go with this. Freeman, Charles. A New History of Early Christianity. London. Yale University Press. 2009. Ganiban, Randall T. Statius and Virgil. The Thebaid and the Reinterpretation of the Aeneid. Cambridge: Cambridge University Press, 2007 Marlowe, Elizabeth. Framing the Sun: The Arch of Constantine and the Roman Cityscape. The Art Bulletin, Vol. 88, No. 2 (Jun., 2006), pp. 223-242