In early years, Islam spread by being an attractive and simple faith, making many converts, especially among those turned off by the requirements of the Christian Church. It also spread by conquest, quickly taking Palestine and Syria, Egypt, and Persia. Much of North Africa and Spain, plus large areas in Asia, followed within 50 or 100 years. In these conquered areas, the Jews and Christians were not required to convert, although pagans had less protection. Non-Muslims were required to pay a poll tax, which encouraged gradual conversions "of convenience"--similarly to how many ambitious people in the late Roman Empire became Christian because it helped their political future.
Later growth was slower, and I suspect much was fostered by Muslim traders, who worked in China, Indonesia, and other areas.