The answer is that they didn't convert all that quickly at all. Those areas that did convert to Islam did so quite gradually. Much of Northern Africa, as former Roman provinces, were Christian during the Arab conquests, but gradually became more Muslim over time, often thanks in part to economic pressures. Similarly, non-Roman city states in East Africa with deep connections to Muslim areas through trade over the Indian Ocean gradually converted to Islam over centuries. It's no mere accident of history that Islam's spread mirrors trade routes: it was easier to find trust and economic security between trading partners when they shared a religion.

However, you are mistaken in saying that Christianity has been slow to spread in Africa. There are hundreds of millions of African Christians thanks to heavy-handed and concerted evangelism efforts. Native religions are much less common. Of course, as with Europe, native beliefs are often incorporated into the way they practice Christianity. Typically, Christianity is more prevalent in areas that never developed strong Muslim communities and received heavy evangelism efforts, while Islam is found spread out along ancient trade routes. This isn't to discount communities like the Christians in Egypt, who have been there since Roman times.

In Islam someone simply had to submit and they were in. Islam means submission in Arabic and a Muslim is simply a believer. If a person in a newly conquered part of 7th century north Africa thew down their idol and approached the warriors saying "I am a believer." They would have to accept that person into the community of believers.

This is polar opposite to many many religions. The barrier to entry in Islam was so low anybody could get in. Unlike other faiths it was not restricted to one ethnicity or language group. Because all of Islam had to study it in Arabic it forced a kind of integration into greater Arab culture. But because Islam had to gain converts form so many diverse peoples it developed different flavors,if you will, as it spread.

So in Islam we have a eastern expansion to all the way to Indonesia. And a western expansion all the way to Spain. Then later beginning in the 14th C. the Turkic expansion into the Balkans which follows a similar pattern to the early movement into Christian Egypt.

In Persia in the 6th century Zoroastrianism was the predominant faith. Zoroastrianism is a religion which warships the supreme god Aurora Mazda, and has very clear good and evil motifs. They took right to Islam and the Ideas proposed in the Abrahamic ideals. The tenements of Zoroastrianism had striking parallels to early Jewish interpretations of the Torah. As an aside, some like Dr. Patricia Crone have theorized a common ancient Sumerian origin for both faiths, Judaism and Zoroastrianism. With the language being similar to Arabic and the cultures similar the transition was very easy.

In north Africa there were mainly Christian kingdoms in Egypt to Morocco. So in this area we did not see the same level of mass conversion as we did in the east. Here Islam was more reserved for the elite military class. This was due to the non pagan nature of the population. The concept of Dhimmi, or people of the book, led to a surprising level of tolerance in the dar al Islam. The people of the book meaning Jews and Christians were not to be hindered directly simply taxed to practice their faith, way better than dying IMO. The Coptic Christian and ancient Jewish population of Medina and Jerusalem were never forcefully converted due to their almost sacred authority over the Bible. Mohammad actually learned everything he knew about God when he went to Medina and engaged in six years debate with the local rabbi's. Mohammad's life is commonly divided into the pre Medina years when he was living in Mecca, which are markedly more pagan, and after in Medina.

This practice and treatment of conquered peoples facilitated trade and culture between the tree faiths in a surprising degree. Warriors picked up cultures and the cultures and flavors of those cultures in their form of Islam as it spread east and west. This is drastically different than Christendom which had the inquisition tantamount to a cultural genocide. Not to mention all of North and South America decimated in the name of god, gold, and glory, Deus Aurum Gloria . That is tree continents cultures homogenized by Christ. Shit forgot Australia, thats a fourth. Islam needs to catch up.... The forceful conversion in Christendom arguably caused a historical nightmare that destroyed Shamanic, Druid and pagan cultures around the world forever. (Thank Valhalla for the Icelandic Sagas saving my Norse heritage! Not all are are so lucky.) Christendom is unsurprisingly the least tolerant of the Abrahamic faiths having more people die in the name of Jesus than any other religion. Sadly as we constantly see in history religions of peace perverted to justify war and conquest.

TLDR: But, it really goes down to early Islam being made up of a professional army of very efficient warriors. Horse archers where the Abrams tank of their day. This allowed for the vast expansion early on. As they expanded we see different levels of conversions. Today we have Bulgaria which had Islamic control for 700 years and is now mostly Christian. At the same time we have Indonesian which is the largest Islamic nation on the planet. With approximately the same amount of time under Islam. It comes down to the initial faith of the conquered peoples. Pagans were more readily converted. Where as Christians and Jews were just taxed as people of the book. I would love some feedback.

I thank the beautiful book Osmon's Dream, /r/C_S_T, Dr. Nicoli Antov, and Dr. Patricia Crone for my inspiration.