Mohamed.
It began in Mecca with Mohamed having "visions" and recruiting people to follow his new thoughts. He was kicked out of Mecca for making trouble. He fled to Medina. There he gathered more followers, often criminals or bandits. After he had gotten enough support he led his followers back to storm Mecca and force their religion on the people there. Mecca is the center of Islam to this day.
The Suni and Shi'a follow different teachers. Immediately after Mohamed's death there was some confusion as to who would lead. Various people came forward to lead. Several different schools of theology popped up and began competing. From two of those schools came those two sects.
Islam spread, throughout most of its history, by force or threats. The same way that Mohamed took Mecca, his followers took other cities. They would threaten and murder to get people to convert. They took over large portions of the world.