Why did the spread of Christianity happen?

Because Jesus wanted it that way; and because his disciples complied with Jesus wishes. Jesus trained his disciples regarding how to spread it (Matthew chapter 10 & Luke Chapter 10)(also Matthew 28:19,20.)

Answer:

It was Jesus' command and commission to His disciples, throughout all the generations to come, to the end of the age of man's rule on earth:

"...Jesus came to them and said, 'All Authority in heaven and on earth has been given to Me. Therefore GO AND MAKE DISCIPLES OF ALL NATIONS, baptizing them in the name of the Father and of the Son and of the Holy Spirit, and teaching them to obey everything I have commanded you. And surely I AM with you always, to the very end of the age.'" (Matt.28:18-20 NIV)

Answer:

From a historical perspective, you could check out the related question, shown below, "Is there any real historical reason why the people of Europe converted to Christianity?"