Islam itself did not spread rapidly after Muhammad's death, as regions such as Egypt would not become majority Muslim until centuries later. The political movement that coalesced around Muhammad did achieve a flurry of major military victories over the next century and I will attempt to explain why this occurred, though this is a very complex question and historians by no means agree with each other on all these points!

The short answer is that the Romans and the Persians were exhausted after twenty five years of war (which only ended roughly four years before Muhammad's death), whilst the Arabs were led by able leaders and had better morale. But before we start, it is worth pointing out how problematic our sources are. There is basically no coherent account surviving from the seventh century narrating the full course of the conquest - the best we have is an Armenian history covering events up to ~660, a work of a limited geographical scope, and an Egyptian chronicle written at the end of the century, which basically talked only about Egypt and began in the middle of the Arab invasion of the region. Some fragmentary material in Syriac exists as well and we can piece together some events from other sources too, such as from the Book of Pontiffs written in Rome, two Spanish chronicles from the eighth century, and of course there is the archaeological evidence, but we definitely do not have a full understanding of exactly what happened. There is also a wealth of Arab sources, but they were written down much later, in the late eighth century at the earliest. There is plenty of invaluable material there, but we have to subject them to the same degree of source analysis that we do for any other source. For example, the size of armies were often wildly exaggerated and martial victories celebrated as superhuman feats by basically every ancient and medieval writer, so there is no reason for us to believe them when Muslim sources report similar things. No source is objective or a mirror of past events. The best we can do is to look at all the sources we have and draw some preliminary conclusions, so I'd be very wary about any firm answers that you may find on the Internet (or in books for that matter).

Take this answer with a grain of salt as well, especially as I deliberately avoided the analysis of specific battles or individuals, none of which can be adequately dealt with by me due to the limitations of my own knowledge. It doesn't help that we don't have enough material to reconstruct specific tactics or motivations - who knows what actually happened in the famous battles we hear so much about in later sources? Instead, I will examine the broader trends within the late-antique world, which I am a bit more confident on, and demonstrate that the rise of Islam was a dynamic and complicated process. Don't let this get you down though, as there is A LOT of research being done on the seventh century (finally) and historians are constantly refining the vague picture we currently have, so there will be something more comprehensive in the future!

So where do we begin? We have to start with the last great war of antiquity. The Roman-Sassanian War of 603-628 was a key event in the history of the Near East, and if it wasn't followed by the Arab conquests, it would have been the defining event of late antiquity. During the war, the Roman administration slashed the wages of soldiers and bureaucrats, melted down 'volunteered' church silver to pay for the war, and the citizens of Constantinople made do without the traditional free bread dole. From the mid-610s onwards, the empire lost its wealthiest provinces in the Levant and Egypt to the Persians, whilst Anatolia was regularly raided. The situation in the Balkans wasn't great either, since the Avars and Slavs, both occasional allies with the Persians, pillaged as far as Greece, with some tribes even settling in the area. Remaining imperial territories in Italy and Spain were attacked by the Lombards and the Visigoths respectively, with Italy also being the hub of a number of internal revolts. The Heraclian dynasty itself had come to power in a brutal civil war in 608-10, a war that directly contributed to the collapse of the Roman Near East in the face of the Persian invasion. Constantinople was besieged in 626 and emperor Heraclius could do nothing to help his capital - he was leading the one functioning army he still had in a risky campaign in the Transcaucasian mountains. The fact that he was personally leading an army was exceptional enough, as no emperor had done that for centuries. Through luck and military skill, Constantinople repelled the Avar besiegers and Heraclius won several miraculous victories when he led his army deep into the Persian heartlands (with a little help from the Turks), causing a crisis within the Persian court that led to the deposition and murder of Khusro II; the new regime in Ctesiphon quickly sued for peace.

Though the Romans 'won' the war, it was a close shave and it easily could have gone the other way. The situation afterwards wasn't much better. The Persian armies in the Near East were only evacuated when Heraclius negotiated a deal with their general, as he couldn't get them out otherwise. Even then, Roman armies were returning to regions which hadn't seen imperial rule for nearly a generation (610s to 628). The Arab confederations that had guarded the desert frontiers were now of doubtful loyalty too, as they may well have been employed by the Persians during the occupation. This is quite important, since both the Romans and the Persians had relied on Arab auxiliaries to defend their frontiers with Arabia. Without a strong rapport between the imperial capitals and their border clients, the two empires were actually quite permeable. Already in the 610s there were records of Arab raiders into Palestine taking advantage of the wider conflict; though we have little direct evidence, a rather ineffective frontier policy must have continued to provide a tempting target for Arab raiders well aware of the two empires' weaknesses.

Within the empire itself, the Romans now also had to deal with the fact that large chunks of the population, miaphysite Christians and Jews in particular, were generally treated quite well by the Persians during their occupation and so they cannot have looked forward to being once again ruled by an empire that saw them as schismatics or worse; the imperial government generally followed the Chalcedonian form of Christianity. There was also the lingering effect of the plague from the 540s and the long-term financial decline of the empire, which can be best seen in the string of military mutinies in the late sixth century. In a very real sense, the Roman empire had expanded too much under Justinian with too little internal reform, and it now faced problems that it was increasingly unable to handle in structural terms.

Not surprisingly, the Persians fared even worse. There was now a civil war between rival claimants after various able candidates for the throne conveniently died. Its armies were therefore uncoordinated and more focused on their internal enemies rather than defending their frontiers. The Persians used to have a confederation of Arabs that defended their border with Arabia too, but the Lakhmids' power was destroyed by Khusro II during his last great destructive war, so the border was now more permeable than ever. According to recent studies, Persia was also relatively decentralised and had very sharp divides between different classes, so the fact that they lasted as long as they did (Yazdegerd III was killed only in 651) is quite impressive in itself.

When Muhammad returned to Mecca in triumph around 630, Heraclius was at height of his power, as he had just returned the True Cross to Jerusalem in a grand procession at the same time, no doubt hoping that he was now able to slowly rebuild his empire. Unfortunately, Heraclius had also issued a number of damaging edicts. Jews were to be forcibly converted across the empire, and Heraclius began to push for the unification of the various Christian churches in the Near East. To bridge the gap between Chalcedonian and miaphysite beliefs, he and the like-minded Patriarch Sergius of Constantinople created the doctrines of monenergism and later monotheletism, compromise Christological formulations that ignored the core issue of how many natures Jesus possessed, but focused instead on how Jesus possessed 'one energy' and 'one will' respectively. Rather than a cynical attempt at bridging different beliefs, it is now seen as the natural evolution of Chalcedonian Christianity, so it shouldn't be a surprise that it worked to a certain extent - quite a few miaphysite leaders and even some 'Nestorian' Christians once again returned to communion with the imperial church. However, it also made some Chalcedonian and miaphysite Christians very angry, since this policy can also be seen as a heretical innovation and it was occasionally imposed with force if people did not accept it willingly (though this was no doubt exaggerated by later sources) - we for instance can point to reports of persecution in Egypt (though also of reconciliation, it's hard to tell what actually was going on there in the 630s) and revolts in North Africa and Italy that can be directly linked to radical anti-monothelete monks. Heraclius' policy, though perhaps well-meaning, in fact did a great deal of damage to the empire.

End of part 1