Why was Jesus' birth set on the 25th of December by Christians?

It was absolutely because of the Pagan solstice celebrations. When Christianity was in its infancy around the end of the 1st century c.e. (not the beginning or middle, but the end, when the gospels were written by whoever wrote them (it was not the apostles, they had all been dead for decades)), it was easier to sell this new religion to non-believers when they could maintain many of the same traditions they had before Christianity.