An important thing to remember here is that these numbers are not accurate reflections of what actually happened. Modern historians of the Arab conquests are reluctant to believe the figures presented by the sources, the vast majority of which were written centuries later and were the results of a later political, cultural, and historiographical context. The earliest sources simply do not provide this information. From a sixth-century source we learn that Emperor Justinian's army, across his entire empire, consisted of 150,000 men - we should be cautious about this number, but that seems about right, as the largest armies recorded by contemporaries were in the tens of thousands. As a result, field armies larger than 20,000 are, I think, rather unrealistic for any point in the seventh century, particularly given the economic and political problems within the empire in the decades leading up to the 630s.

It is probable that there was a military engagement at or near Yarmuk, but the evidence we have is patchy. From c.660, the Armenian historian Pseudo-Sebeos recorded only an engagement of a 70,000-men strong army that was defeated by an Arab ambush, but the Romans only suffered 2,000 casualties. This may or may not be the genesis of later stories concerning the Battle of Yarmuk. At around the same time in distant Burgundy, the well-informed chronicler Pseudo-Fredegar noted that there were 150,000 Roman mercenaries against 200,000 Arab soldiers, both ludicrous figures. Yet the Burgundian historian also noted something rather interesting, that during the night before the battle 52,000 Romans died because they were smote by the Lord, which can be read as an indication that the plague struck the army and led to mass casualties - the plague was certainly active in the region and must have hindered operations even if there wasn't such an inconvenient mass outbreak. In the eighth century, the best source is perhaps Theophilus of Edessa, a court astronomer at the Abbasid court and whose now-lost history probably formed the core of a number of late chronicles in different languages, such as the work of Theophanes the Confessor (Greek), Michael the Syrian (Syriac), and Agapius (Arabic). In this telling, the Roman army consisted of 40,000 and they all died in the battle (which is of course highly unlikely). Another factor was raised however, as it appears that some Roman soldiers proclaimed one of their generals, Baanes/Vahan, as emperor, which could not have helped Roman morale or cohesion in the battle shortly afterwards.

I am less familiar with the Arabic sources, but I believe that similar differences can be detected in their accounts - a natural result of them being only written down in the late eighth century or even later. The sizes of the armies are not consistent, no more or less than the sources I've mentioned so far, so it is difficult to even establish the broad outline of what happened. These accounts are also affected by the addition of certain literary tropes, with the most extreme being the idea that the Roman soldiers were chained together to prevent them from fleeing; this is not attested in any Roman/non-Roman sources before this, but according to some Arab chroniclers this was the modus operandi of both the Romans and the Persians. This is surely an indication of their attitudes towards the two imperial powers, which they saw as decadent polities ripe for the conquest. It makes for a good story, but they cannot be taken as accurate depictions of what actually happened.

In reality, we simply don't know all that much about Yarmuk. I'm sure there was a major engagement, that the Romans (probably) outnumbered the Arabs, but beyond that we have to speculate. Perhaps the Roman forces were plagued by the.... plague and they may have faced internal unrest, yet for the actual battle itself we are completely blind. Pseudo-Sebeos' battle consisted of an Arab ambush, and according to the dependents of Theophilus, there was dust blowing into the eyes of the Romans. That's it, that's literally all the tactical information that we have from the earliest sources. I would not so easily dismiss the information preserved in Arab accounts, but there remains a big gap between historians working on different sources and I think that a better understanding will only emerge when we have a comprehensive overview of each source, their context, and their reliability. If we favour the early sources, how do we account for later Arab narratives? If we prefer Arab sources, how do we then explain the words of Pseudo-Sebeos, Pseudo-Fredegar, and Theophilus? In my view, to understand the Arab conquests, historians have to understand and study sources in Arabic, Syriac, and Greek together, but that is still pretty rare. Robert Hoyland, one of the few historians who follow this approach already, says only the following for the Battle of Yarmuk in his recent book on the conquests:

Chroniclers now focus on an encounter that, in retrospect, was viewed as turning the tide against Byzantium and in favor of the Arabs: the Battle of Yarmuk. The name refers to the river that flows westward along the modern border between Jordan and Syria, and empties into the river Jordan, just south of the Sea of Galilee. The Arabs were led by two of their most famous generals: the aforementioned Khalid ibn al-Walid and Abu ‘Ubayda ibn al-Jarrah, who comes across as a tough but fair and pragmatic man. Both were from the prophet’s tribe of Quraysh, but whereas the former was from a clan that long opposed Muhammad, Abu ‘Ubayda had been a close companion of the prophet from the very start of his mission. The emperor Heraclius had traveled to northern Syria to obtain better intelligence about events and realized that matters were serious, and so he appointed his top general in the east, the Armenian Vahan, to take charge of the operation. Heraclius was particularly worried that the key city of Damascus was under threat, and so he dispatched messengers to instruct legions with spare capacity to release some troops for the defense of the Syrian capital. Vahan marched from Antioch with the main army, apparently routing a small Arab exploratory force on the way, in the vicinity of Homs. In the early summer of 636 “an enormous multitude of Saracens set out from Arabia and headed for the region of Damascus”; presumably the success of the initial Arab raids had encouraged many others to join the venture. Worried by this, Vahan wrote to Theodore, a patrician of Edessa and a senior financial administrator, to bring reinforcements, and he arrived with a further 10,000 men. Once all were assembled, they encamped together, right by the bank of the river Yarmuk.

The Byzantine force looked very strong, but fortune was not on their side. An initial encounter in July 636 resulted in a defeat for the contingent of Theodore. This led to an argument between the latter and the Armenians, who subsequently proclaimed their own general emperor while abjuring Heraclius. Theodore’s men withdrew and the Arabs seized the moment to attack; some had been hidden in ambush around the Byzantine camp and they now jumped out and fell upon the enemy. The Byzantines attempted to flee, but the dense mud of the flood plain bogged them down and the heat of the sun overwhelmed them. Many thousands were either cut down by the sword or slipped down the steep sides of the river valley and were either crushed or drowned. The exact details of the battle are difficult to recover, but the loss of Byzantine life was evidently considerable, for it sent shock waves around the empire, even as far away as Gaul, where one Frankish chronicler spread news of the calamity. The latter and another Latin source blame at least some of the loss of Byzantine life on an outbreak of the plague, which was raging in Syria at that time.

As for why the Arabs won, I think it is important to remember that some Arabs were integrated part of the Roman world already and that Roman sources regularly report them as fighting on behalf of the Roman/Persian empires. We should not see them as unsophisticated, but instead as members of the world of late antiquity as well, most evident in the Ghassanid and Lakhmid confederations that thrived in the imperial borderland. Their power was broken by the end of the sixth century and we know little about the smaller Arab groups that took their place due to the paucity of sources, but I think that it is unlikely that we can dismiss the continuous participation of (some) Arabs in imperial politics. In my view, Arab 'raiders' were not nomads alien to the imperial powers, as the core of their forces must have been veterans well-versed in Roman/Persian strategies and the landscape they were battling in. Again from Hoyland:

From this perspective the Arab conquests began as an Arab insurrection, that is, the early conquerors were not invaders coming from outside the empire but insiders trying to seize a share of the power and wealth of the Byzantine state. This helps explain why the Arab conquests were not particularly destructive: the leadership already had close acquaintance with the empires and they wanted to rule it themselves, not destroy it.

There were many, many other factors behind the successes of the Arab conquests and I attempted to summarise them here, but note that despite its length it is still not comprehensive. More research is being done all the time on this topic and I am confident in the near future we can further rewrite the conventional narrative of the Arab conquests, so watch this space!