Presumably it spread through the early church for the same reasons it is shown to have spread in Scripture. When miracles will be sufficient testimony for a people, God does miracles.
Some non-historical thoughts:
The message itself is appealing: God loves you. He came to earth to pay for all your wrongdoings by dying and now you are cleansed from all offense against him, and he now sees (and on Judgment Day will see) you as holy, righteous, and good. There will be no more pain, and so forth. These things are not presented in Scripture as hopes (in something that may or may not happen), but things that
will happen.
The Word has power that can affect people in ways unlike anything else, such as triggering a recognition of Truth, rather than convicting someone that an explanation is logical. Although there are various nonspiritual ways to explain why the Ethiopian was quick to believe (he was in Jerusalem because he already had some faith), the bottom line was that he was given a basic understanding of some things from the Word and the good news and he believed (narrative in Acts 8:26-40).
I've heard that God has ensured there remain avenues in every culture through which people can connect to the truth of the gospel. I don't know if that's true, but wherever it was the case, Jesus would have been an answer that rang true to a people group where there was a tension (or pain) about something (e.g., the resurrection story for people desperate to know they have eternal life, instead of a reason to
hope for it).
The quote from Celsus, referred to in JackRT's post, belies a lack of recognition of the uniqueness of the gospel. e.g., where is the part for salvation where I have to pay something to God for my sins? This is contrary to fallen human nature. We're wired to not be satisfied with a justice where there is no punishment to the lawbreaker. When a person (whose faith is not cold) learns the good news he gets an injection of God's glory into his heart (which hopefully will bring him to Life
).