I think there's two parts to this that are worth addressing:

The first is: how specifically did "Islam [begin] in the middle-east in 620-630, [and then] go all the way through the Iberian Peninsula and into France in roughly 100 years?"

The second is: was this "expand[ing] so quickly"?

I'll answer the second question first although it's not a particular areas of expertise for me. Were these conquests "so quick?" Well, yes and no. There is no doubt that the Muslim conquests, in particular, were extraordinary in world-historical terms for the speed and size of their expansion. But a large part of that is knowing A: the odds they faced in defeating both the Byzantine and Sassanian empires and B: how small and obscure the Arabs had been in historical terms before that.

I'll discuss both of those points a bit further down, but what I mean by this is that in absolute terms the ability of an antique or late antique military to simply conquer the area extending from France to the Middle East in the span of 100 years is not particularly remarkable. The Visigoths had done it. Justinian had done it. And so on.

In other words, it's not really what they did, but who did it and how they did it that strikes me at least as being the more salient issue.

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So going back to the first question: how specifically did "Islam [begin] in the middle-east in 620-630, [and then] go all the way through the Iberian Peninsula and into France in roughly 100 years?"

Well, unfortunately we're not 100% sure. Documentary sources from this time period (i.e. written texts written within living memory of the events in question) are extremely limited. The later material recorded by "early" Arab historians (i.e. writing 200 years later) like Tabari is, while often extensive, is nonetheless of questionable reliability. Moreover the material specifically relating to the conquests of North Africa and Spain are a tiny fraction of that literature. In one of the best such sources, Ibn abd al-Hakam's Conquest of Egypt [and North Africa and Spain] the relevant bits on the conquest of Spain, as I understand it, amount practically to a few pages.

The basic answer however is that having established a polity predicated in large part on outward expansion, the Arab armies were able to defeat the Byzantines and Sassanians and beyond that expanded in such a way that they met little resistance and that made it difficult for them to be pressed back.

So in the case of the Byzantine territories, after the Battle of Yarmouk in 636, the Arab armies were practically free to expand into North Africa as they pleased, meeting some resistance in Egypt but nothing particularly capable of halting their advance. Byzantine efforts to retake cities like Alexandria after the initial conquests failed, partly because of the military capability and skills of the Arabs, partly for logistical reasons, and partly because the Arabs, perhaps unexpectedly, they started contesting control of the Mediterranean with ships of their own almost immediately.

The westward advance after Egypt does not appear to have met particularly difficult resistance. Some berber tribes, moreover, appear to have been amenable to Islam and converted quite quickly and apparently in some instances en masse. This further bolstered the ranks of the conquering armies such that by the time the Arabs invade Spain their army is no longer strictly "Arab" but is actually largely berber.

Spain, moreover, as ruled by the Visigoths was not a very well organized state enterprise and was unable to pose significant resistance, to my understanding.

I think it would be beyond the scope of this post the sort of how they did that portion is one that's not easily settled. It's all well and good to say that "Muhammad established an Arab polity bent on conquest" but what did that look like this? This is hotly debated. If you read someone like Fred Donner he'll emphasize the religious (albeit not in the earliest years necessarily yet "Islamic") component. Robert Hoyland, not seeing evidence for strong religious institutions in the early text, emphasizes the Arab-ness of the conquests. Still others debate the extent to which this early Islamic state was "Islamic" or even "a state." Lack of contemporary documentary sources make it difficult to make absolute conclusions, though I personally lean towards Hoyland's writings, if only perhaps because I find "fanatical motivation towards apocalyptically minded Jihad" or whatever to be difficult to quantify strategically even if it does bear mentioning.

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The last bit that I think would be worth discussing is the Battle of Tours/Poitiers. While this features extensively in European writings as a decisive battle without which Christendom would have fallen, I think that's actually been pretty well rejected by the current consensus of historians who write about this.

In regards to Tours itself, although the Arab army was, numerically, apparently "normal" for an Arab expeditionary force (although the numbers involved in battles is always a great point of debate), it doesn't seem to have been a particularly well-organized expedition and it wasn't regarded as being of special strategic importance. The Arab historians barely seem to notice it and they didn't try to mount any other major such expeditions into France again as far as I'm aware.

Moreover, it seems pretty clear to me at least that the Ummayad empire (as the Arab empire now was under the Ummayad dynasty) had reached its territorial maximum. The relevant comparison that I think supports that would be the Battle of Talas (751) fought on the other side of the empire where the Arabs soundly defeat a Chinese force but are unable to translate that into significant territorial expansion into China. Moreover in 750 the Abbasid Revolution permanently fractures the Arab empire, with the remnants of the Ummayads fleeing to Spain, and although Islamdom as a whole would continue to expand the trend from 750 onwards was the splintering of the empire into smaller independent or satellite kingdoms that weren't capable of making pronounced gains in the face of well-organized opposition in Western Europe or against the Byzantines.

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Source wise:

Robert Hoyland's In God's Path is a great and short book about the conquests.

Fred Donner's Muhammad and the Believers or his earlier book, partly outdated but nonetheless extremely good The Early Islamic Conquests

There's also two lovely podcasts from In Our Time on both Tours and Talas that are worth a listen:

http://www.bbc.co.uk/programmes/b03pm7dv http://www.bbc.co.uk/programmes/b04kf8ps

Both of these feature Hugh Kennedy, whose books on the conquests are also excellent, though I prefer Hoyland and Donner.

edit: phrasing of a sentence.