My sources are my classes on PaleoChristian and Islamic archaeology, and my personal interest in mythology, but I'm certainly no specialist on the subject.

For Christianity, as you know Christianity first spread through the Roman Empire. Christianity was quite popular for its worldview, which appealed to the poor, and the numerous slaves present in the Roman Empire. Christianity gives a prominent place to the poor, saying they are loved by God, whereas pagan gods aren't necessarily benevolent. Christianity also has a much clearer explanation of the afterlife, and has a stricter set of rules to follow, which is comforting. In contrast, the Pagan religions seemed to be for the elites. When the first Christians were persecuted, that also brought them some sympathy. Basically Christianity was more appealing. It's also highly proselyte, whereas while the Romans did romanize other people through their religion, it was more about absorbing conquered people's gods and claiming they're equivalents of Roman gods, thus you get a religion which isn't very centralized and clear. When the Roman Emperor Christianized, it was thus the state religion, and considering its tenants, Christianity was much less tolerant of "false Gods". With the backing of the state, the old pagan shrines and idols were destroyed and defaced, and replaced by Christian churches. When the barbarians invaded the Roman Empire, Rome was fully Christianized, and the barbarian kings typically first kept their old pagan beliefs, but for political reasons, to make alliances or to rule over the conquered people, they converted to Christianity, until becoming warriors for Christianity. Clovis and Charlemagne are quite notable examples of this, the latter forcefully converted or massacred other pagan Germanic people. Later on, the Vikings and Balts were converted in similar ways.

The spread of Islam is somewhat comparable. Mohammed was surely very charismatic and had a large following, with which he managed to defeat and conquer neighbouring pagan Arab tribes. The Arabs were eventually united under Islam, also a proselyte religion, and as their initial conquests were fantastically rapid, it convinced people of the truth of Islam.