It was not peaceful, but also very very far from pure conquest by fire and sword. The first thing to note is that Islam had the great good fortune of erupting out of Arabia shortly after a debilitating, 20+ year war between the Roman (Byzantine) and Persian Empires, which the Romans won. Persia fell rapidly, and the Arabs conquered the Middle East up into Anatolia.

The second thing is that Egypt and North Africa had significantly different versions of Christianity than the Orthodox version aggressively pushed by Constantinople (they were Monophysite). Relations were not good, with various emperors veering between tolerance, and attempting to force one, Orthodox Christianity on the entire empire.

When the Arab armies came along, in some cases they were actually aided by the locals, because they had a very tolerant policy. In terms of religion, they gave the conquered peoples 3 choices: Conversion, Tribute, or the Sword. In other words, conversion was preferred, or if not then you could pay tribute (which was often less than taxes to the Byzantine Empire). They gave their subjects freedom of religion within those constraints, which was a better deal than they got under the empire. However, if you didn't care to do either of those, then you were put to death. Needless to say, the great majority of people chose to pay tribute and keep their religion, which was a better deal than they got before. Over time of course, the majority of the population converted to Islam, but this process took centuries. To this day almost 10% of the population of Egypt is Coptic Christian, which is Monophysite.

It is my understanding that as word of these practices grew, conquests along North Africa got easier and easier. They got more difficult again in Spain, where orthodox (Catholic) Christianity predominated.

My main source (Karen Armstrong, A Brief History of Islam) paints a relatively benign picture of the Islamic conquests, perhaps too gentle a picture. However, the Muslims were certainly far more tolerant with their conquered peoples than say the Crusaders.

Quoting Paul Freeman of Yales' lecture:

And finally, number four, the attitude of the conquerors and the conquered. The attitude of the conquerors was what Peter Brown in the reading for Wednesday will call, "a garden protected by our spears". This is a quote from one of the conquerors. The Arab conquerors considered these to be wonderful civilizations that they were not going to pillage or destroy, but rather protect. "A garden protected by our spears." But they were planning on enjoying the garden, not merely standing on the outside defending it for other people to enjoy.

The conquerors were confident in their religion, so confident that they didn't need for others to recognize it or convert. It also gave them the confidence to accept new ideas from Greek civilization, from Persian civilization, from India