This is a complex question that would require a book to answer fully, as there were multiple factors involved, and the salient points have already been noted, so I'll try my best to cover the military aspects that lead to Arab supremacy in the battlefield.
First, there's the nature of nomadic warfare, which, outside of the rare standing battles, was mostly raiding other tribes for booty. The nature of this type of warfare required highly mobile troops and skilled hit-and-run tactics, and the ability to flee towards the inhospitable desert when pursued. The Arabs had many, many years to perfect this type of warfare. Their horses and camels were bred for this exact type of warfare, and their weapons and armor evolved to suite.
An early Muslim commander of the Rashidun caliphate, Khaled bin Alwaleed, exploited this strength in mobility and leveraged it against the slow-moving forces of the Persian and Byzantine armies, who had evolved to fight defensively, in semi-static blocks, sometimes even chaining the troops together to deter cavalry charges. Both were heavily dependent on infantry, who had to march to the battle and needed extensive baggage trains for supplies. The Arabs, on the other hand, were all mounted, on either horses or camels. Their infantry rode along and had the added advantage of being able to mount up and become mobile-infantry mid-battle, allowing them a great deal of maneuverability. The whole army was capable of cutting through terrain that a marching army would find all but impossible, allowing the Muslim forces to show up whenever, wherever, and choose the battle-fields that worked best for them, all the while, the enemy had to march their troops through the heat and race to cutoff the Muslim forces, which left their own forces haggard, dehydrated, and at a tactical disadvantage when they showed up... Only to find that the Muslims had set up camp around the only source of water within miles.
I mentioned Khaled bin Alwaleed above, and I suggest you read up about him if you're interested in this era. He was a military genius who was really the Rashidun's main asset. Their were others of course, but in my humble opinion, and the opinion of his contemporaries, he was key to Muslim victory over the Sassanids in particular, and was instrumental in laying out the strategy that beat back the Byzantine's counter-offensive under Heracles. He truly was exceptional.
I've already written a tome, so I'll try to be brief in regards to the Arab's armament and morale. The former is, I think, not given enough consideration, the latter is well covered.
The Muslims were lightly armored with chain mail when they could afford it, but used layered cloth mostly, which was surprisingly effective, particularly against projectiles. Their ranged weapons were short bows and javelins, their infantry melee weapons were Nubian spears and straight swords, of which, the Indian variety was highly coveted for its strength ( the traditional scimitar was handed down much later from the Byzantine cavalry saber). The mounted troops charged with two-handed lances, allowing pinpoint accuracy and multiple uses, and when in the clinch, they drew their swords. But their greatest weapons were the horse and the camel.
The Arabian breed is the perfect light-cavalry mount. Small enough to be highly maneuverable on the battlefield, aggressive enough to not shy away, and is overwhelmingly superior in terms of stamina, allowing the Arab cavalryman to charge many times before having to change mounts. The camel was a logistical dream, as it can go for three days without drinking water, and more than that without any sustenance. It can carry a man and all his supplies, and if they ran out of food, they could slaughter a few and feed many. On the battlefield, the camel spooked horses, and allowed for a high platform from which to shoot from, allowing the Muslims to fire from right behind their front line and down into the enemy's midst. And as I've mentioned earlier, allowed for any reserve infantry to quickly maneuver and take advantage of sudden weaknesses, and in the case of an enemy route, the ability to chase down a good many of the survivors.
To summarize; the Arabs used every advantage they had, and exploited every limitation and weakness that the enemy possessed, and coupled with exceptional leadership and high-morale, the Arabs managed to beat enemies, Byzantine and Sassanid, who's military "culture" and doctrines had evolved to fight a certain way.
I'm on mobile, so I'll post my resources in a few hours.
Talking about the development of Islam in Arabia is tricky, since we basically only have two vaguely contemporary sources - the Qur'an, which is a collection of Muhammad's teachings and seems to have been written down and compiled by the 640s, and the Constitution of Medina, a set of rules governing Muhammad's fledgling community in Medina that is preserved in a late eighth-century biography of the Prophet (but is probably from much earlier due to its archaic Arabic). What do the two sources tell us? Not very much I'm afraid. Muhammad definitely preached for a religious movement of some sort against the status quo, one that made some people (presumably at Mecca) angry and this led to some attacks on him and his followers. Then there was a migration to Medina, which led to further fighting both internally and externally between Muhammad and his enemies.
This might be surprising, but the Qur'an is actually pretty bad at giving us narrative details about what happened in these critical years. There are very few mentions of Muhammad and geographical details are sparse too, so this vague picture is the best we can extract from the Qur'an. For all the other details of Muhammad's career at this time we would have to rely on later sources, particularly from the Sira of Ibn Ishaq written in the late eighth century, which only survives in an edited form by Ibn Hisham and al-Tabari from the ninth century. I don't think there is any reason to question the general framework of the story - that Muhammad fled from Mecca to Medina and then fought to impose his rule on both his home town and on the wider world. But the details, such as the precise details about who did what to whom and the motivations behind their actions? I'm not so sure. Oral history is a fantastic resource and early Muslim scholars were very diligent in compiling their history books, often discarding many less credible stories from their collections, but we still have to be careful about trusting their evidence implicitly. Even contemporary sources are terrible at recording what actually happened, so we need to treat them the same way.
It is possible for instance to make the case that the general gist of the story from the Qur'an was expanded in later centuries by writers writing in a very different context, who were unfamiliar with the seventh-century world and thus left an imprint of their ninth-century mindset in their texts, subtly changing the narrative so that it made sense to them in order to explain away the sometimes contradictory stories found in the Qur'an. This did not have to be malicious in nature, nor is this evidence of cynical attempts to change history (though this obviously happened, I doubt anyone here is that naive!), but it should at least be obvious that no historian can ever represent the past completely accurately and that all narratives are shaped by their own circumstances. My comments here represent my own perspective shaped by my own experiences, and is thus similarly inadequate for describing the complex nature of the rise of Islam. This sceptical approach is however probably the best we can do for now, though if you are interested in further details the New Cambridge History of Islam (2010) has a very good chapter on this. Take a look too at Michael Cook's Muhammad (1983) for a basic introduction to the criticisms against the traditional narrative. It is also worth noting that this is not an attitude aimed only at the history of Islam. No academic historians nowadays would take at face-value the narrative provided by any ancient or medieval writer any more; though the results are now often less certain, it does make everything much more exciting, as we are now exploring the world beyond the text, one that was for various reasons obscured by the primary source itself.
For the second part of the question, I think my answer linked by /u/farquier did a good job in pointing out the problems historians have with writing about this period. Nonetheless, I did attempt to explain how the Arab Conquests happened and why it was successful, though of course I am always open to further questions and suggestions from others :)
I recommend /u/shlin28's posts in this thread:
While this doesn't go into the early history of Islam in Arabia( a hotly contested topic due to the extreme paucity of evidence on the issue) it does discuss the early Islamic conquests in some detail and outlines some reasons for its success. The general takeaway is that Islamic forces(or, to tweak /u/shlin28 a bit, the "community of believers") was able to spread in large part because of the collapse by exhaustion of the Byzantines and Persians in the face of constant frontier warfare and internal religious dissent, in part because they were largely unobjectionable in actual governance, and in a very large part because they were very much like other seminomadic peoples in other situations who lived on the peripheries of empires and were able to attack size control of those empires in times of crisis when the normal system of tributary relationships, buffer statedom, and border raiding broke down.