Where does it say December 25th is Christ's birthday?

It doesn't and it isn't. If you work through The Bible following all of the laws of sacrifice set up in the book of Deuteronomy. It actually shows that Jesus could be born on no other date than October 2, 7 bc . They were expecting him, that is how the 3 kings knew to go to Jesus. And why king Herod tried to kill him

ANOTHER OPINION:

December 25th was a pagan festival commemorating the birthday of a false deity. The church intentionally chose this date to to establish a rival celebration.

AnswerIn addition to the opinion above, the Roman festival of Saturnalia occurred at or around the winter solstice and extended into the new year. When Rome became officially Christian it was appropriate that the new festival commemorating Christ's birth (Christ's mass) should supercede the old false pagan festivals and not sit alongside them, or else people would adopt both.

Therefore Dec 25 was accepted as the day on which Christ's birth and, more importantly, his incarnation, should be celebrated. What is certain is that it would not have been then, as this was in the middle of winter. And in Luke, we read that 'shepherds were abiding in the fields, keeping watch over their floocks by night'. This would never have happened in the cold harsh winters that hit the mountainous areas of Israel. So Jesus would have to have been born in warmer weather.

Though the date isn't important, the commemoration of his birth and incarnation is important, as, next to the crucifixion and resurrection this is the most important event in the history of humanity, for just as you cannot have the incarnation without the crucifixion and resurrection, you cannot have these without the miraculous incarnation of Christ - an event that needs to be commemorated for all time.