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.