There are a number of reasons why Islam spread around the world, and this spread began even before the death of Prophet Muhammad.
Islam spread quickly because of the appeal that this set of beliefs offered, as an alternative to Polytheism. The first Islamic communities, or caliphates, were established soon after the death of the Prophet, and due to the success of their military operations, both on land and at sea, these caliphates grew quickly.
While Jews and the early Christians treated Islam with suspicion and derision, the Islamic way of life, and the prosperity that came with it, was greatly appealing to the nomadic people of North Africa.
By the time 100 years had passed since the death of the Prophet, the message of Islam had spread, by means of battle, across large parts of Europe, Asia, and Africa.