This is nearly 1000 years worth of history featuring 20 or so Islamic Empires who were often fighting each other.
See
Discussion of Lepanto:
http://www.bbc.co.uk/programmes/b06nrqv8
Early expansion in an oversimplified nutshell:
Mostly Christian Arab tribes had provided significant numbers of soldiers for both the [Eastern] Roman and Persian Empires for centuries. Romans and Persians had fought a series of destructive wars where Persia captured then lost significant ME territories.
Both Empires had also been badly affected by plague further weakening them and severely affecting the tax base necessary for military upkeep. Reduced populations also meant that fewer potential soldiers were available.
Numerous Arab groups had been raiding the borderlands in 6th/7thC and likely started to notice that these Empires were struggling to respond. Lacking money to pay Arab mercenaries likely resulted in many of them defecting and turning to raiding.
As the Arab groups got more united they faced what was left of the main field army of each Empire in battle and defeated them. In pre-modern times losing one battle could lose you a whole Empire as you no longer have a viable army to defend the territory (the next closest Roman Army would have been months away in the Balkans, and was needed there anyway).
Due to geography, both Empire lacked natural defences against attacks from Arabia, so after the main field army had been defeated, there was little they could do to defend huge areas of land and capitulation and tribute was the only real option.
While it is sometimes presented as something truly miraculous that 'outsiders' like the Arabs could defeat 2 mighty 'superpowers', this is not really accurate. The conquests were impressive, but far from unique.
Both Empires were severely weakened and a shadow of their former selves. Arabs had been serving in the Empire's armies for a long time so were well versed in tactics and organisation. While the Arab forces were likely outnumbered, it probably wasn't by all that much and anyway, raw numbers are not anywhere as important as other factors such as experience and morale which the Arabs had the advantage in. There are examples in history of armies winning when being outnumbered between by more than 10:1 and many when that number is 4:1 or so. Even if the Arabs were outnumbered 2:1 (might have been less) it would not be particularly unusual for them to have won.
They did conquer vast territories in a short space of time which, while impressive, doesn't match up to the Mongol Conquests