The most salient feature in the dramatic rise of Islam was the military conquests of Arabian Muslims. Muhammad set the tone by attacking caravans of pagan Arabs, whose rejection of the prophet, and Islam, forced Muhammad to flee from Mecca to Medina in 622 AD. By the time of his death 10 years later, Muhammad had conquered Mecca and most of the pagan tribes of Western Arabia. He recruited soldiers in part through promising glory for their martyrdom, forced his captors to accept Islam, and systematically destroyed their pagan temples and religious artifacts.
After Muhammad's death in 632, his first four successors, or caliphs, continued military campaigns that quickly captured most of the middle east and spread into Europe. Though the victors did not force their new subjects to convert to Islam, they required them to pay a special tax to be able to practice any different "Abrahamic" faith (e.g., Christianity or Judaism). Since the majority of the conquering armies were already Islamic, there was also considerable social pressure to convert to the new regime's religion. Moreover, it is instructive to note that the Qur'an (Islamic bible) contains several passages urging Muslims to ultimately "fight" and "slay" non-believers.
Military conquest, of course, was not the only major factor in the spread of Islam. Traders passing through Arabia heard of the new religion, and helped spread it to their homelands. And, as with the growth of Christianity, pilgrims set out from Arabia to carry the word to peoples throughout the Mediterranean world. One could presume that there was an inherent appeal in the message of Islam that attracted a devout following. Perhaps, in an historically unstable and warring area of the world, having powerful conquerors who provided relative stability inspired confidence in their new leaders, and eventually the religion they brought with them.