The short answer is "nobody knows", but that does not a blog post make. Also, two shows I watched this week ("Good Eats" and "Big Bang Theory") both proposed that Christmas is a recent invention by the (by implication evil) Church to try to suppress Saturnalia, a pagan holiday.
The problem (if you want to consider it a problem) is that Jesus' birth date (and in fact birth dates in general) was not recorded in any official document. Biblical accounts give a description of the year in very vague terms, and say nothing about the day of the year. About all we "know" is that it took place during the reign of Herod. The exact years of Herod's reign are a matter of historical debate as well. I'll say more about that in a future blog post. And so a "line in the sand" was drawn and the best information at the time was used to determine the year when the modern calendar was invented.
Why choose December 25th to celebrate the birth of Christ, of we don't know the original day? That's a much simpler question. Recent (like last 300 years or so) scholars have proposed all sorts of reasons, but I think we can apply Occam's Razor and come up with a much more plausible reason for picking that date.
You see, in the early Church, Christmas was not celebrated. That Christ was born was certainly acknowledged as fact, but the important aspect was not that He was born but that He became man. Since it was always recognized by the Church that life begins at conception, the important Holy day to celebrate was Jesus' conception, the feast of the Annunciation. On that day, the angel appeared to Mary and she consented to bear a son conceived by the power of the Holy Spirit.
Of course that date was not recorded either, so what day should the Church fathers celebrate it on? There are several lines of reasoning that all lead to picking one particular day. For one thing, we are told in the bible narrative that Mary went to visit her cousin Elizabeth, who six months pregnant with her child, John the Baptist. Luke's Gospel further tells us how the angel Gabriel announced John the Baptist's conception to his father, Zechariah, when Zechariah was serving as high priest on the Day of Atonement (in September), around the autumnal equinox. Therefore, Jesus' conception would have been around the spring equinox. Also, even without this evidence, if you wanted to pick an "auspicious" time for Jesus' conception, the natural time to pick would be around the spring equinox, when the growth of life all around us would be a symbolic reminder of the Christ child growing in the womb of His mother.
For whatever reason, the date of the Annunciation was set in the very early Church to be March 25th, and that was the day it was celebrated, even to the present time. Add nine months, and you come up with Jesus' birth being on December 25th. The first record of Christmas (the feast of the Nativity) being celebrated was in 336 AD, nearly 300 years later. Saturnalia (December 17-23) certainly was a Roman feast around that time, and so was Sol Invictus (December 25). Was Christmas "moved" to coincide with those feasts, or is Christmas a "continuation" of those feasts? Certainly not, I would claim. When the date of the Annunciation was set there was no knowledge that Christ's birth would also be celebrated 300 years later, so the choice of December 25th is coincidence.
Now, was the celebration of Christmas used to replace these feast days? Perhaps. It was a common practice of the early Church to incorporate customs and traditions of other cultures into the Church's tradition. The intent was to make it easier for converts, by using symbols familiar to them, rather than forcing them to abandon family customs and their own culture. Finding an existing feast day that corresponded with a Roman feast day would no doubt be easier for Roman converts than making them change their routine and calendar to correspond to a different set of feasts. In that sense, the popularity of Christmas today may be due to those earlier pagan feasts, since prior to Rome embracing Christianity, the feast of the Nativity was not considered an "important" feast day (like Easter or the Annunciation).
However, to say that Christians are "celebrating" Saturnalia is false. we are celebrating the birth of our Lord and Savior, Jesus Christ. Of course, falling on the same day as Sol Invictus (the "Unconquered Sun") is kind of a cool coincidence, since Christ is our light.