Like most civilizations throughout history, the Islamic way of life spread through military conquest. Due to decisive military strategy, Islamic civilization achieved many military victories both within and without the Arabian peninsula. Unlike many conquerors throughout history, the Muslim conquerors were often liberating nations from comparatively wicked despots, and presenting a life and faith that were compelling and progressive in comparison to what they had suffered prior.
Islamic civilization also spread due to mission work. Once military conquest had established a caliphate, or Islamic state, missionaries would mingle with local populations to propagate the teachings of Islam. People were excited about Islam, as they had not been about religion in some time, evidenced by the fact that even traders and people traveling for economic reasons had a hand in spreading the faith.
The Islamic Empire spread through the conquest of territories both in Arabia and outside of it. An effective military and an efficient management system allowed for the empire to grow.
The primary factor in the spread of the empire was their ability to conquer through the military. The empire benefitted from attacking areas suffering from despotic leadership. This created internal strife, which made the conquest much easier. One specific example was Spain: once conquered, the people of Spain were treated very generously. This was a welcomed change compared to the harsh leadership they had suffered. These actions helped win the people over with Islam.
Another specific reason the empire grew, outside of the conquests, was mass conversion to Islam. When the military conquered a territory, it did not force the people to abandon their culture. Rather, the empire installed Muslim leaders that worked with the locals. What they found was the people converted on their own so that they could be closer to their leaders.
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.