In the early seventh century, a merchant trader named Muhammad received revelations from the Angel Gabriel about the final word of God. According to the Islamic faith, born out of this event, Muhammad became the final prophet of Islam, and it was his mission to spread the faith. Muhammad began by spreading his faith in Mecca. After a longer story, involving a flight to Medina, gathering followers, and having to claim Mecca for the Islamic faith, Muhammad is successful in creating a base for Islam in Mecca.

Prior to Islam, Mecca was a trading city that engaged with a lot of monotheistic faiths but remained polytheistic. One trend we see in world history is that it is easier to convert a population away from polytheism towards monotheism, but it is harder to convert from one monotheistic faith to another (on a large scale). Islam was the first successful monotheistic religion to take hold in Mecca and the larger Arabian Peninsula.

Through the Umayyad Dynasty, the empire that formed after Muhammad's death, originally headed by a loyal follower named Abu Bakr, Islam spread throughout the old Persian lands to the East and North Africa (and into Spain!) to the West. While sometimes this spread was militant, Islam mostly spread through peaceful means. For example, people living in Islamic territories were granted religious tolerance. However, non-Muslims who practiced Christianity or Judaism (called dhimmi, or "people of the book") were charged a tax that non-Muslims did not have to pay, called the jizya. This encouraged non-Muslims to convert.

Conversion also brought a better economic relationship with the Muslim World, who at this point in history (600 CE–1450 CE) were the most successful traders on the Silk Roads and Indian Ocean. One example of this is in the West African Sahelian Kingdom of Mali, whose King Sundiata converted to Islam, along with some of the African elites. Later Malian Kings, like Mansa Musa, went on the pilgrimage (hajj) to Mecca, spreading faith and African goods along the way. Converting to Islam connected African Kings to the Trans-Saharan Trade network, where Muslims in North Africa were interested in buying Sahelian products like gold and salt.

We also see mass peaceful conversion in Southeast Asia, particularly in port cities like Malacca. Sufi mystics, Islamic missionaries who used fused Qur'anic faith with local practices, converted many Hindu and Buddhist traders due to the promise of equality and community that comes with Islam. Low-caste Hindus especially appreciated the concept that all followers were equal under the eyes of Allah, something they did not experience with the highly restrictive caste system. Traders in SE Asian port cities converted to Islam to engage in trade and to experience a more egalitarian religion. Today, Indonesia has the highest percentage of Muslims of any country in the world, including Islam's homeland, Saudi Arabia.

What factors contributed to the rapid spread of Islam in the 7th and 8th centuries?

Islam began in the Arabian Peninsula, but it spread all the way to Spain and India, touching many places in between, within just a few hundred years. Much of this was due to the political instability of this area. European rulers had grabbed land across this area but had not effectively organized the people they supposedly ruled, and many of these people felt no real sense of loyalty to their supposed rulers. In the seventh century, Arab Muslims began to conquer these territories in the Sasanian and Byzantine empires, quickly extending their influence across three continents. Because these areas had been long exasperated with their political instabilities, they didn't have any strong oppositions to their new Muslim rulers, who did not rule purely from a religious standpoint.

Once political and cultural stability was a bit more certain, the Umayyad Dynasty (which came a bit later) had a foundation from which to spread Arabic and Islamic culture, including religion. The Umayyads did not seek to convert its subjects outright, but non-Muslims did have to pay a special tax, which then did finance the efforts of further spreading their political (and, indirectly, religious) expansion efforts.

When we are trying to account for phenomena that have to do with religion, we have to be careful in our answers.  To people who believe in a religion, the answers may be different than for those who do not believe.  For example, a Muslim might argue that Islam spread rapidly because its teachings were clearly true and therefore touched the hearts of those who heard them.  Historians, by contrast, do not accept the truth of any given religion and do not accept arguments based on the idea of a true religion.

Historians would say that one reason that Islam could spread was that it had done a very good job of uniting all of the Arabs.  The Arabs had previously expended a great deal of energy fighting one another.  Now, they directed the energy outward.  They were also inspired and motivated by the idea that they were spreading the true faith to other people.  Some historians argue that they were also helped by the fact that the Byzantines and Persians had been at war with one another.  This weakened them both and left them vulnerable to the Arabs.  As Arab armies conquered, they brought their new faith with them.

A second factor that helps to account for the spread of Islam is trade.  Arabia was connected to many other places through trade routes.  This allowed the ideas of Islam to spread rapidly to other places.  This process was also bolstered by the practice of having Muslims come on pilgrimage to Mecca.  There, Muslims from various areas made contact with one another and spread their trade networks even more widely.

Thus, we can say that Islam spread because of the military might of the Arabs and their widespread trading networks.