This is a fascinating question, and one that I appreciate because many people don't realize that North Africa was Christian for centuries before the Islamic conquest of the region. For example, probably the most important single theologian for the whole Western Christian tradition is Augustine of Hippo, who was born, raised, and spent the majority of his life in modern-day Algeria and Tunisia; ethnically, his mother was Berber. And of course, Alexandria, Egypt was a vitally important center of Christianity in the early Church, with a patriarch that exercised influence across the Roman Empire and many prominent theologians. How did this hotbed of Christianity for nearly five centuries change so quickly and overwhelmed to Islam? One answer is that the traditional narrative is somewhat overstated, and indeed Christianity did survive in pockets throughout the Middle Ages in North Africa, and not only in Egypt. Still, it's hard to deny that a large part of the population did accept not only Islamic rule but Islamic faith.

One possible contributing factor is the pre-Islamic division within the North African Church: the first major one was the Donatist schism, in which a group split from the rest of the Church over the issue of whether or not believers who recanted their faith under Roman persecution could be restored to the fellowship of believers. In the case of priests and bishops, the question also applied to the validity of their official acts such as baptism or ordination. The Donatist church was still a live issue at the end of the 4th century, when Augustine wrote against them, and that division continued in some form or another until the Muslim conquest. Adding to that, Vandals invaded in the early 5th century, and they brought with them an adherence to Arian Christianity, which differed primarily by rejecting the idea of Jesus' full divinity. There were still some deep-seated divisions in the religious landscape, even before Islam appeared, and as can be seen in many historical contexts a house divided falls more easily. The Arian belief in particular may have found more common ground with Islam than the others; some scholars (Charles Speel, to name one) have argued that Arian conversions to Islam were much more common than from Trinitarian Christians.

Another element that scholars previously pointed toward, but I believe has somewhat fallen out of fashion, is the influence of monasticism. Particularly, it was thought that the strong monastic foundation of the Egyptian church helped it weather Islam, as opposed to western North Africa whose monastic communities were relatively few and weak.

Culturally, it's debated as to how deep the Christian roots really were among the native North African populations. There are prominent Berber Christians like Augustine, and many in Egypt, but Augustine also notes that even in his own day Christianity was more urban, with country people more inclined toward traditional pagan worship. That's what "pagan" originally meant, pertaining to the countryside; in its early centuries, Christianity was a remarkably urban phenomenon. Even when the Berbers did accept Christianity, they didn't necessarily accept Roman/Byzantine culture along with it, keeping cultural links at an arm's length which in other areas helped solidify religious bonds. When Islam came, and the Arab culture with it, they didn't feel such strong ancestral links to the rest of the Greco-Roman Mediterranean world.

So by the Islamic conquests of the 7th century, a given people may conceivably have gone from pagan to Catholic to Donatist to Arian in the previous three hundred years; another shift to another flavor of monotheistic faith may not have seemed such a big deal to some. And of course, we cannot ignore the social pressures: conversion to Islam would bring economic and social benefits, relief from the jizya tax and opportunities for higher social standing and service in government. Between the pre-existing religious divisions and the definite social, economic, and political advantages to conversion, Islam proved an appealing option for many 7th and 8th century North Africans.