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.