How did the birth of Christ come to be celebrated December 25?

There are several hypothesis. The most commonly held is that Christians co-opted the Roman feast of the return of the Sun, Saturnalia and other mid-winter feasts,

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According to this theory, early Christians deliberately chose these dates to encourage the spread of Christmas and Christianity throughout the Roman world: If Christmas looked like a pagan holiday, more pagans would be open to both the holiday and the God whose birth it celebrated.

Despite its popularity today, this theory of Christmas’s origins has its problems. It is not found in any ancient Christian writings, for one thing. Christian authors of the time do note a connection between the solstice and Jesus’ birth: The church father Ambrose (c. 339–397), for example, described Christ as the true sun, who outshone the fallen gods of the old order. But early Christian writers never hint at any recent calendrical engineering; they clearly don’t think the date was chosen by the church. Rather they see the coincidence as a providential sign, as natural proof that God had selected Jesus over the false pagan gods.

Prior to Constantine the persecuted Christians separated themselves from any identification with their pagan neighbors and were not likely to be "baptizing" pagan feast. This did occur post Constantine but there is some evidence that north African Christian sect, the Donatists, had marked the birth of Christ as December 25.

Another possible reason that December 25 was chosen may be more theological than physical. This was to link the conception of the Christ child with the date of the crucifixion. The month and date of the crucifixion is approximately know as it is tied to the Jewish feast of Passover, 9 months from passover is late December to early January.

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Around 200 C.E. Tertullian of Carthage reported the calculation that the 14th of Nisan (the day of the crucifixion according to the Gospel of John) in the year Jesus diedc was equivalent to March 25 in the Roman (solar) calendar.9 March 25 is, of course, nine months before December 25; it was later recognized as the Feast of the Annunciation—the commemoration of Jesus’ conception.10 Thus, Jesus was believed to have been conceived and crucified on the same day of the year. Exactly nine months later, Jesus was born, on December 25.d

This idea appears in an anonymous Christian treatise titled On Solstices and Equinoxes, which appears to come from fourth-century North Africa. The treatise states: “Therefore our Lord was conceived on the eighth of the kalends of April in the month of March [March 25], which is the day of the passion of the Lord and of his conception. For on that day he was conceived on the same he suffered.”11 Based on this, the treatise dates Jesus’ birth to the winter solstice.

So is there a definitive answer to the questions surrounding when Christians celebrate the birth of Christ. Not really. It is like all study of times and peoples ancient. There is a lot of conjecture based on limited facts and the things they left behind. What was the real purpose of Stone Henge and the other stone monoliths and stone circles dotting the European countryside? What was the purpose of the great pyramids in Egypt- were they merely monuments to massive egos? What about the pyramids of Central America, why that shape?

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In the end we are left with a question: How did December 25 become Christmas? We cannot be entirely sure. Elements of the festival that developed from the fourth century until modern times may well derive from pagan traditions. Yet the actual date might really derive more from Judaism—from Jesus’ death at Passover, and from the rabbinic notion that great things might be expected, again and again, at the same time of the year—than from paganism. Then again, in this notion of cycles and the return of God’s redemption, we may perhaps also be touching upon something that the pagan Romans who celebrated Sol Invictus, and many other peoples since, would have understood and claimed for their own, too

http://www.biblicalarchaeology.org/daily/biblical-topics/new-testament/how-december-25-became-christmas/