The short answer is that the Byzantines and the Persians were exhausted after twenty six 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. This is also a very big topic, so although I'll try to cover all the important aspects, there's always room for others to improve on this answer!

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 Muslim 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 Byzantine-Sassanian War of 602-628 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 Byzantine 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. Western Byzantine 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 Byzantine 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 Caucasus 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 and led to the deposition and murder of Khusro II; the new regime in Ctesiphon quickly sued for peace.

Though the Byzantines '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, Byzantine armies were returning to regions which hadn't seen Byzantine 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, whilst large chunks of the population, miaphysite Christians and Jews in particular, were generally treated quite well by the Persians and so 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 decline of the empire - in a very real sense, the Byzantine Empire had expanded too much in the sixth century with too little internal reform, and it was now facing problems 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 astonishing.

When Muhammad returned to Mecca in triumph around 630 (or maybe not if you follow James Howard-Johnston's argument), 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 monoenergism 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 wider Byzantine 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 persecutions in Egypt and revolts in North Africa and Italy that can be directly linked to radical anti-monothelete monks. Heraclius' policy, though perhaps well-meaning, did a great deal of damage to the empire.

A couple of overlapping dynamics, the proverbial perfect storm if you will.

1.) Most of the Middle East prior to Islam was in the domains of 2 perennial powers: The Byzantine Empire (Eastern Roman Empire) and the Sassanids (Persian Empire). Literally prior to The Arabs erupting out from the peninsula, the two juggernauts had just settled a massive conflict. This war, which took place during the life of Muhammad ﷺ and actually has a Surah dedicated to it, was devastating for both sides. Both were left exhausted and reeling from the losses, neither made any meaningful gains as the result was a return to status quo. Especially vulnerable, should something unexpected happen. But really, who could threaten these 2 superpowers other than each other?

2.) Well as history shows, the Arabs. Prior to Islam, this peoples had been since the dawn of history a scattered nation divided into many tribes. Often too busy warring amongst themselves to form a cohesive polity. Never had they had a unified state, but this changed with Islam. At the time of Muhammad ﷺ's death, the Peninsula had been fashioned into a united body for the first time in history. The take away here is, he had brought the tribes under one faith- Islam, and one common identity that superseded tribal affiliations- Muslims. United and with a powerful new found ethos under this common banner, they were able to coordinate like never before as well.

Additionally, even prior to Islam there were many Arab client states subservient to the Persians and Romans. These periphery players readily defected to the new Arab overlords as they came forth. Both Romans and Persians relied on these Arabs for trade and other services which disappeared basically over night.

3.) A final and arguably the most important dynamic was the Arab leadership. Both in Political and Military terms, the Arabs were blessed with leaders who reigned supreme. On the political front, the concept of Caliphate ensured a sense of strong political hierarchy which is crucial for any Empire. On top of this, the string of 4 successors known as the Rashidun Caliphs were all particularly competent leaders.

On the military front the Muslims had some very, very talented generals. The most obvious one being Khalid ibn Al Waleed, who is considered up there with Genghis and Alexander as one of the eminent military minds of history. His tactics and leadership saw victory time and time again. At which point success just led to more success. The early Muslims had several generals of similar caliber. This tandem of adept political and military leadership proved, as history attests, unstoppable.

Also, on a more specific note about the military tactics, the sheer mobility of the Arab forces proved to be simply too much for the standing imperial armies to handle. As is often discussed here, the imperial armies in question were indeed in a 'weakened and exhausted state'. But too often I see this fact get a tad bit exaggerated, as if this is the only reason the Muslims were able to win. I reject that notion simply on the grounds that not only were the Muslims virtually always outnumbered in the initial phases of conquest, but on top of that were at times engaged on multiple fronts by multiple belligerents. To chalk that off as simply the the Arabs getting 'lucky' is a severe discredit to the actual military prowess of the Arab forces which successfully employed highly complex maneuvers and mental tactics (such as 1 on 1 battles between Muslim commanders and opposition commanders which decimated Byzantine and Persian officers corps'). This topic alone, of the early Arab military tactics, is worthy of several paragraphs and post itself, but for sake of brevity we can reduce this to the advantages of superior mobile tactics exploiting the weaknesses of a standing army. We can see this for example at Yarmouk were Khalid Ibn Al Walid kept moving his forces in and retreating, in and retreating. This forcing the Byzantines to also move their forces in accommodation. In due time, the standing army got throughly exhausted. Precisely at which point Khalid Ibn Al Walid went in for the Kill. He and his fellow generals throughly exploited the advantages of their superior mobility to achieve victory against superior forces (on paper). Ironically enough, the same exact dynamic (superior mobility vs standing army) would lead to the demise of Muslim civilization(s) at the hands of the Mongols some 5 centuries later or so. But I digress.

So basically, how could the Caliphate expand so ridiculously fast? A combination of these dynamics: Weakened opposition, a powerful and united national ethos, and exceptionally skilled leadership on all fronts. The perfect storm.

This is a very topical overview of course and by no means comprehensive, and I'm not basing it off any one source. But if you are interested in further reading, I highly recommend 'The Great Arab conquests' by Hugh Kennedy as a very accessible starting point.

Edit: Just thought I would throw in this fun but relevant fact: Khalid Ibn al-Walid was dismissed at the zenith of his career, for misappropriation of state funds (he allegedly had a perfumed bath on the Caliphate's coffer), and because people were beginning to attribute the success of the Muslims to him rather than God. I believe this is why it is said he was 'undefeated'. He never had that big loss, that India campaign or Waterloo. He just kinda abruptly went off the map, due to devices beyond his control.

A large reason was because the Byzantines and Persians had fought eachother to death for centuries, not to mention the Justinian plague that wiped out almost a quarter of Byzantiums population just roughly 60 years earlier (I'm on my phone so I can't provide depth but I will later if I gett the chance). They were both the only two who could stop them, and their power was depleted and governments corrupt, as Byzantium went through a rough revolutionary period in the early 7th century, by the time Heraclius could recover the empire and defeat Persia the Arabs were on the move. Another reason is Arabs were commonly used by both aforementioned parties as mercenaries very often so a lot of these guys had pretty good experience. Finally, I'd say a large factor played into the hands of Khalid ibn Al-Walid a companion of Mohammad's who was one of the very few generals to never lose a recorded battle, against the Arabs (before he was with Mohammad he fought against him) and against other foreign powers. Several of his crushing victories over the major powers, enhanced by religious fervor, gave huge swaths of land to the Arabs, who would assimilate the local populace.