1. Islam spread quickly due to its military might, strategy, and large, extremely cohesive armies. Islam spread quickly because its philosophy was compelling, progressive, and was considered extremely 'modern' for its time. Universal equality? No class system? No contradictory Gods? Yes, please.
2. These two effects provided a positive feedback to each other. Islam was religion, law system, patriotism, and spirituality all rolled up into a very convenient package that was neatly contained in one book. It was awe-inspiring and lucrative, to say the least.
To draw a rough comparison, look at the United States of America of today. Back in the 1930s, it was just another neck of the woods, reeling under a terrible depression. But by the end of WW-II, it became a superpower. In around 70 years, 'American' products, art, thoughts, values and ideas have percolated through most of the world. It is a kind of soft-power, spread through Hollywood, TV shows, and Voice of America/CNN. Civil rights, animal rights, environmentalism, innovation etc. all took root in America and were exported to the rest of the world. In most of the modern world, the youth want more freedom, liberalism (e.g., the Arab Spring) and less authority, all thanks to the example set by the US.
Islam spread in a similar manner. First, Islamic armies came in and totally pwned reigning empires. Next thing you know, they built giant, imposing mosques from where the azaan could be heard at great distances. Knowing Arabic (later, Persian) got you places, just like English does now. You naturally began to drift towards the cultural mannerisms of the victors, just like people today take up American mannerisms (only my two cents) to 'fit in'. Accepting Islam was pretty 'cool' back then. They had the power, the success, the charm almost, to draw followers.
