Why did the Europeans spread Christianity?

The message spread through people and it spread because that is what Jesus told people to do:
Matthew 28:19 Go therefore and make disciples of all the nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit,

On the day of Pentecost Peter preached the gospel message to people who were gathered in Jerusalem. (see Acts chapter 2 )

These people were Jews and Jewish converts from all over the known world. Many believed the gospel message and became Christians. When they went back to their home countries no doubt they told others, and so the message of salvation through Jesus Christ was spread.

Christians throughout ages have done the same thing