Generally, Roman religion was falling apart. But the 2dn century, many people had lost faith in the ancient polytheistic system. Many religions were spreading (mostly monotheistic or spiritualist religions), but Christianity was on the forefront of all. Christianity had been persecuted at times in the previous centuries; persecutions mostly depended on whatever emperor or governors were reigning. For example, Nero and Diocletian were harsh against Christians, but other Emperors less.

Anyways, around 300 AD Christianity had reached almost all regions of the Empire, despite still being a minority religion. Basically every city had a Christian community, however little. People joined for a couple of reasons: 1) Generally, old religions were failing in providing spiritual assistance 2) People found comfort in the idea of a happy afterlife, unlike that of paganism 3) Christian missionaries were amazingly proactive and solid believers, while pagan priests hold their office more as a job 4) Christian doctrine of peace, love of neighbor, and monotheism was something that appealed to many but was not present in any previous religion 5) Christianity abridged class distinction since everyone was equal in the afterlife; this was very appealing in a society with huge numbers of slaves, women and other people none or less rights.

Christianity was illegal and often persecuted, but martyrs who refused to renounce their faith only showed people how convinced and pure-hearted Christians were. Christianity was legalized in 313 AD with the edict of Milan by Constantine the Great. much has been written about him, and I won't delve too much on Constantine; but in synthesis some say he genuinely believed in Christianity (although we aren't totally sure he was baptized), other think he legalized it because Rome needed a new state religion and he found Christianity suitable. it is important to notice that he gave Christianity a finalized structure with the council of Nicea.

From that moment on, the rise of Christianity was unstoppable. Finally legal, Christians preached openly and the numbers rose to the point that Christianity became the majority religion. In 381 emperor Theodosius made Christianity the official religion of the Empire, and in the successive decades paganism was abolished and outlawed. Temples were tore down and rituals suppressed, but there never was a full scale persecution of pagans themselves. Eventually, paganism (that survived in rural areas) disappeared completely.

This is a rough generalization, but hope it helps!