How did Islam spread successfully in the middle east?

There's bits and pieces of answers in the other comments here, but I'll try to give it more of a synthetic go. During the lifetime of Muhammad and into the 630s, most conversions probably came through direct contact with the Prophet or one of his Companions, but this only accounts for a very small sliver of the Middle East. At the time of the Prophet's death, Islam was still highly localized to Mecca, Medina, and the oasis of Ta’if.

In the next two decades, the Arab conquests rapidly exploded out of western Arabia, and Islam spread with them. Note that I call these campaigns (630s-50s) Arab and not Islamic conquests, and in fact, they might even better be referred to as an Arab insurrection against Byzantine and Persian meddling in Arabia. As these superpowers fell into decline in the 620s, the followers of Muhammad were one group among many of "barbarian" coalitions vying to fill this vacuum of power, and it's clear that Muhammad and his successors would often ally with non-Islamic tribes who could have been attracted to join the conquest for any number of reasons.

Muhammad's coalition had unexpected success, however, and they were able to secure leadership of the Arab tribes as a whole. As they threw off Byzantine rule in Syria and Persian rule in Iraq and Iran, these Arab conquerors settled apart in garrison towns. At this point, many of the non-Muslim Arabs probably began converting to Islam, perhaps encouraged by the divine favor that Muslim armies seemed to enjoy and also by the bonds that they had developed with their Muslims comrades through the hardship and sufferings of war.

Only later as the Arabs increasingly converted to Islam did the purpose of conquest become more about spreading the faith. It seems that during the early conquest of the Middle East, Arab armies gave their enemies a choice between fighting or paying taxes (part of why it looks more like insurrection than conquest). Later, as Islamic scholars wrangled about legal precedents established by the Quran and the Prophet, Arab armies began to offer the additional option of converting and not paying taxes. During the conquests of the Middle East, however, it seems like the assumption was simply that the (non-Muslim) conquered paid taxes to the (Muslim and non-Muslim) conquerors.

So the situation in the Middle East around 650 was one of small garrison towns of conquerors, perhaps totaling 250,000-300,000 Muslim Arabs, among some 25-30 million conquered people, i.e. Muslims made up 1% of the total population. Muslim men were often in the position to take multiple wives and/or concubines, which meant that Muslim birthrates quickly exceeded non-Muslims. Also, as the Islamic conquests spread out of the Middle East in the late 600s, prisoners-of-war started flowing in. Many of these slaves converted to Islam and were freed by their masters. It was considered a pious act to free a slave, especially a Muslim slave, and freedmen were expected to be servants bound by ties of loyalty to the master who had freed them, which also gave the freed slaves certain guarantees of patronage, so everybody won out. Although it's difficult to know whether these slaves actually experienced a conversion experience, the joint practice of conversion/manumission meant that an increasing number of people in the Middle East were considered Muslim.

It's important to note that many of the most prominent Muslims who defined Middle Eastern Islam in the 700s had either been captives or descended from captives. One former captive (Muqatil ibn Sulayman) wrote the first commentary on the Quran. The grandson of another captive wrote the first biography of Muhammad (Ibn Ishaq). And the grandson of another captive founded one of the most influential schools of Islamic law (Abu Hanifa). So it seems like slave converts and especially their (freed) descendants were critical factors in the definition and spread of Islam.

Historians have made a lot out of the motivation to convert to Islam to get tax breaks (which was certainly a thing later on) or to secure a higher social status or even to adopt a religion in a bid for local political power, but these examples at least point to more spiritual motivations. Since there wasn't much coercion, Islam spread slowly. Muslims probably did not secure a majority in the Middle East until the tenth century, and despite occasional and often localized efforts to enforce a single religion, the Middle East as a whole has remained a highly cosmopolitan and multi-religious region right down to the present day.

Robert G. Hoyland, In God’s Path: The Arab Conquests and the Creation of an Islamic Empire (Oxford: Oxford University Press, 2015). - (Absolutely the best source out there; I reread it before writing this post.)

Hugh Kennedy, The Great Arab Conquests: How the Spread of Islam Changed the World We Live In (Philadelphia: Da Capo Press, 2007). - (Not quite as tendentious as Hoyland, and perhaps more thorough.)

Fred M. Donner, Muhammad and the Believers: At the Origins of Islam (Cambridge: Harvard University Press, 2010). - (Good for thinking about what Islam meant before 750.)

Richard W. Bulliet, Islam: The View from the Edge (New York: Columbia University Press, 1994). - (Another great source for thinking about how Islam was defined and spread, although a bit of a later focus.)

Richard W. Bulliet, Conversion to Islam in the Medieval Period: An Essay in Quantitative History (Cambridge: Harvard University Press, 1979). - (Examines conversion especially in Iran based on a survey of biographical dictionaries; argues that most notable people converted within about three generations of conquest.)