How did islam spread so quickly?
In 610 CE, Muhammad (The soon to be prophet) was on one of his visits to a cave. There, an angel appeared to him, and told him to declare. After that, Muhammad traveled around to spread the words he was told to recite stated there is one god, Allah. He began his ministry in Mecca. To be quite frank, Islam spread because Mecca was a extremely busy trade route, and when people passed through the city, they would hear the word of Muhammad.
When Muhammad would proclaim the word to other people, when they would travel back to their cities, they would talk about what they learned to others.