Answer:
Arabs were able to spread Islam due to a number of factors. It is notable that although there are a few occurrences of Muslims carrying out the "Convert or Die" scenario, this is not the most common reason for Islam's expansion both in terms of territory and belief.
1) Arabia - What's that?: Prior to Mohammed, Arabia had been seen as a cultural and political backwater barely worth notice and therefore Mohammed's consolidation of the Arab Tribes was not seriously considered by the Byzantine and Sassanian Empires. This lead to a very porous southern defense and miscalculation of the Islamic Armies.
2) Byzantine-Sassanian Wars: Since the Byzantines and Sassanians considered each other the only real civilizations in the region, they fought each other numerous times in the early 600s C.E. over control of what is today East Turkey and Iraq. These were intense religious wars (Christian vs. Zoroastrian) and greatly weakened both empires at just the wrong time.
3) Tribal Command Structure: Arab Tribes and warriors owed their loyalty to their Sheikhs and leaders without fault because the Sheikh provided each person with all of the amenities and protection necessary to live a full life. Whenever the Arabs went to war, the Sheikhs could count on the ability to muster their tribesmen as soldiers without issue.
4) Support of the Locals: Since the Byzantine Empire and Sassanian Empire were engaged in religious zealotry, it stands to reason that they did not tolerate their religious minorities very well. The Muslims professed to (and did) treat minorities far better than either of these Empires. This led to religious minorities (especially Jews) from within the conquered territories helping Muslims to secure outposts and positions, freeing the Muslims to continue to spread the religion.
5) Citizenship by Conversion: The Islamic Empire was interesting in that being the leader or a person of import in the Empire was not contingent on ethnicity or place of birth as was the case of most prior empires. All a person had to do to gain power and prestige was to convert to Islam. This simple integration mechanism resulted in massive number of Persians, Berbers, and Turks converting to Islam. In fact, the furthest expansions of Islam (Spain, India, West China, South Russia, the Balkans) were all accomplished by non-Arabs (respectively Berbers, Persians/Mughals, Turkmen, Azeris, and Ottomans).
6) Warfare: The leaders charged with expanding the influence of Islam were thoughtful and diligent military leaders. Caliph 'Omar, who conquered much of the Levant, Egypt, Persia, and Iraq was well-known as a strategist and a fighter. It was this knowledge of warfare that allowed Muslim forces to expand the territory's influence.
7) Universality: Islam, as a religion, claims salvation for all people in all places, making the religion very attractive, especially to people born into low castes and bad economic positions.
8) Commerce: Islam, like most religions, spread also through peaceful means of commerce and education whereby non-Muslims learned of the faith and adopted as their own amidst business dealings with Muslims.