Why did Christianity become so widely accepted?

Because it was growing more and more popular and more people were becoming Christians.

Christianity became so widely accepted because it developed a strong institutional structure compared to other religions, which has made it prevail over time. It is also due to the fact that through Christianity there was a profound transformation of the Roman Empire that currently represents the most important religion in the world, since the Roman authorities took on the task of making it an official religion, which entailed to its expansion throughout Europe, as well as in other continents such as Africa, Asia, Oceania and America.

It should be remembered that Christianity is based mainly on the Belief of God as Supreme Being, who sent his only begotten son Jesus, for the salvation of Humanity.