The empires of Byzantium and Persia had been fighting over the territory between them since about 570. By 630, many of cities on the front had suffered destruction of walls and crops, and many of the men of fighting age had been drafted for one of the armies or otherwise killed. The area of the front was generally called Syria, covering some of what we now call Iraq, as well as Jordan and Palestine.
It was under Byzantine control when Caliph Umar set out to attack. Between 634 and 636, the Arab army conquered Damascus and won some pitched battles that left all of Syria in their control. At the same time, another army headed into Persia and conquered its capital, Ctesiphon.
The Byzantine bishop of Jerusalem, Patriarch Sophronius, surrendered after a long siege, in 637. Arabs conquered Gaza, and while negotiating a one year cease-fire with Byzantium, incidentally took Antioch. Damascus, Antioch and Jerusalem were the chief cities of Syria, and were the original centers of Christianity. As soon as the truce expired, they finished occupying all of the Holy Land, Mesopotamia, and much of Egypt. Alexandria fell in 642, and the Byzantine Empire was pretty much over.
Umar was the second Caliph, and he died in 644. He had led his armies in all this conquest, but he hadn’t directly participated in the fighting. His focus was on political structure. He kept it minimal, setting up a taxation system and appointing a governor, but otherwise leaving each locality to do things its own ways. His first priority for tax money was to give the other Companions of Mohammed a pension so that they could focus on organizing the Koran and hadiths. His death ended the political unity of Islam, though. More on that later. Conquest continued, under various generals.
Byzantium had controlled North Africa, which was now open for rapid conquest. It was only loosely governed to begin with, and the inhabitants had never felt much allegiance to Constantinople. This was a problem everywhere; Byzantine taxes had been high, and the Byzantine Orthodox church had suppressed local religious sects, Jew and Christian. It wasn’t immediately obvious that new masters would be worse, and they may have been better at first. Muslim histories record the Christian and Jewish residents coming to them freely and swearing allegiance, promising to help dislodge Byzantine rule.
Once Byzantine troops in Tripoli and Carthage were disposed of, the residents didn’t resist conquest at all. In North Africa, the Berber tribes welcomed Islam and joined their armies.
Sicily was invaded in 652; Crete and Cyprus were Arab by 653. Muslims now held much of the former breadbasket of Byzantium, resulting in food shortages there.
By the time the Companions of Mohammed were splitting into factions, they were receiving tribute and taxes from most of the ancient world: Syria, Palestine, Persia, Egypt, Arabia, and North Africa. A lot of money and power was at stake. The Companions agreed that they’d keep power within their own tribe, the Quraysh. But they couldn’t agree on which individual: Ali, Mohammed’s cousin and son-in-law, or Uthman, another influential man. And from this dispute the world has never quite recovered.