The most historically accurate reason for why these regions became Islamic is that, over time, the people came to accept the ways of the politically dominant group in their society. Contrary to what people like /u/DeadMenTattleNoTales often suggest from a very historically ignorant perspective, this didn't happen at the tip of a sword but at the pressure of social and political policy.
I mean honestly, if you have a cursory knowledge of history and a little bit of common sense, you can see how little sense the 'convert or die' theory makes.
Muslims killed all the inifdels
How is this even reasonably possible? The population of Egypt or greater Persia numbered hundreds of thousands if not millions at the time of conquest. How many heads is a state logically gonna chop off before they lose any and all legitimacy over those they seek to rule and tax. Doesn't make any logical sense.
Muslims have forced everyone to convert to Islam
Using Egypt and Persia again as examples, if this is the case, that they were forced to convert, given that it took over 8 hundred years for Egypt and Persia to become the 90%+ Muslim they are today- I must say these politically incompetent murderous Arabs did a really shitty job at forced conversions. Except, you know, thats not the case. In Fact, the Umayyads at one point actively discouraged conversions because they were making too much money off the Jizya. Kinda throws a wrench in that theory.
non-Muslims have fleed because of implementation of Sharia
This explains that mass exodus and migration across the the Middle East and North Africa that we saw in the 7th century, as the citizens of these lands from the Iberian peninsula to the Punjab scrambled to flee the Arab overlords. Wait what? There is no historical record of such a thing? Impossible.
As you can see, each and every single one of these bullet points is horse-shit and screams massive ignorance of the history of this period. But who am I if not a random internet commentator? My word doesn't mean jack. So let me leave you with words from two of the foremost historians in the field of Early Islamic Conquests.
[Speaking on the conquest of Iraq] In field battles at Qādisiya and Jalūlā, and cities like Ctesiphon at Tustar, the Sasanian forces were decisively defeated. With the collapse of the Persian army, the Arabs were prepared to make fairly easy terms with the rest of the population - they did not massacre townspeople and villagers, they did not seize their houses or their lands, they did not interfere with their religions and customs, they did not even settle among them. They demanded only that taxes be paid and that the people did not aid their enemies. Whether the taxes were higher or lower than they had been under the previous administration we cannot tell, but we can be certain that most people in Iraq thought that it was a bargain well worth making.
And
Hugh Kennedy, The Great Arab Conquests - Chapter 3 In many ways acceptance of Muslim rule was the result of Muslim policy towards the enemy: it was almost always preferable to surrender to the invaders and to make terms and pay the taxes than to resist to the last. The Islamization and Arabization that followed conquest over the next two or three centuries would not have occurred if political conquest had not already succeeded, but they were not a direct and inevitable consequence of that conquest. Instead, it was a gradual, almost entirely peaceful result of the fact that more and more people wanted to identify with and participate in the dominant culture of their time. Hugh Kennedy, The Great Arab Conquests - Chapter 12
And
As we have repeatedly seen, the Muslim conquerors put little or no pressure on the recently subjected populations to convert to Islam. Any attempt at compulsory conversion would probably have provoked widespread outrage and open hostility. As it was, the Muslim authorities established working relationships with the heads of the churches and other religious institutions that were now in their power. Conversion when it came was partly the result of fiscal pressures, the desire to escape the hated poll tax, but also because conversion provided an opportunity to escape from existing social constraints and to become a part of the new ruling class. Being a Muslim had always been essential for anyone who wanted a career in the military. By the tenth century, and before in some areas, it had become very difficult to have a successful career in the civil bureaucracy without becoming a Muslim. Attraction, not coercion, was the key to the appeal of the new faith.
And
Hugh Kennedy, The Early Arab Conquests - Chapter 12 According to the ecological thesis, the early Caliphs were merely riding the tiger of the expansion of the Arab peoples, over which they had little real control, at least at the outset. It is for this reason that proponents of the ecological thesis often prefer to speak of the “Arab conquests”, rather than the “Islamic conquests.” The view that the conquests were essentially more “Arab” than “Islamic” was partly rooted in the observation of an undeniable fact, that the conquests were not carried out primarily to secure the religious conversion to Islam of the conquered populations, at least beyond the Arabian peninsula. For, as is well known, the conquerors were content to collect tribute from non-Muslim religious communities outside Arabia that tendered their submission, and to leave them free to continue in their former faiths. Fred Donner, The Byzantine and Early Islamic Near East - Chapter 7
Im not asking you to accept any particular narrative, just to use your head and at least try to be objective.
credit to /u/caeser10022 for the sources.