The way this is determined is that in Luke 1:5 it says John the Baptist's father was a priest in the Abijah division. Further in Luke it says that an angel came to John's father while he was serving in the temple that his wife will get pregnant.

Also in Luke, it says in the 6th month of John's mother's pregnancy Mary conceived.

When you look into when the Abijah priests served (in Chronicles) and convert it to our calendar instead of the Hebrew calendar you can calculate and end up at the end of December.

I don't have all the math, but you can calculate it if you want to, but there is a decent reason to think it was the 25th or thereabouts.