Like most civilizations, Islamic civilizations expanded their empire through conquest and effective leadership. The Muslim world had several characteristics that aided its expansion.
1) Location, location, location. Because Islam began in the Arabian Peninsula, which is at a crossroads of Asia, Europe, and Africa, the empire naturally expanded into each of these regions. Many of the nearby civilizations, such as the Byzantine, were weak and unorganized at the time, so Muslim leaders were able to enter and overtake these competing empires and continue with their expansion. Islam gained followers, often supporters who disliked their current lives and leadership and hoped to have access to a new way of life under Arab rule. Fun fact: Islam does not allow leaders to force conversion on others, so most conversion to Islam was completely the free will of the conquered people.
2) The Religion. When Muhammad died, the Muslim religion was a little unclear on who should be leader and how succession worked in general. As a result, there were splits in the religion both in terms of beliefs, such as Sunni and Shi'a, and in terms of leadership. Different caliphates (or ruling families) that rose to power over several hundred years moved the "headquarters" of Islam around according to their own needs and interests. That's how centers of Islam developed in places like southern Spain.
3) Muslim culture. The Muslim expansion into vastly different areas meant both a spread of Muslim learning and interests and a blending of Muslim learning with other cultures. This helped to bolster the civilization because it brought great discoveries to different parts of the world by combining ancient knowledge from Greeks, Romans, etc. with new discoveries by Muslims.