Firstly, the only accounts of Jesus' life were written after his death by people who couldn't have been alive when he was. The health back then simply wouldn't allow it.
Second, this is no evidence for Christianity at all. You may as well be asking how any other religion spread. For example, you don't believe in Vishnu, but Hinduism was a very dominant religion in many parts of the world. If Vishnu doesn't exist, then how did so many people get to believe in him so quickly?
Do you see what's going on here? I'm asking the same question, but with a different god that you don't believe in. This applies to every religion. Religion is passed down from generation to generation.
People in power grab hold of the religion, and force other people to believe in it, or they condemned, and sometimes even hurt people who do not believe. People involved in religion convince others through trust (Remember, this is a time where literacy wasn't even very common. You had to trust what someone said). Parents would have many children, and then convince each child about this. Because the children trust their parents, they are helpless.
For these and many more reasons, religion, even if it's wrong, spreads fast.