The reason is that December 25 was, at the time this tradition began, the shortest day of the year. There are four days which the ancients recognized as significant: The shortest day of the year, which is now December 22, the longest day of the year, which is now June 22, and the two days in which the day and night are exactly equal in length, which are March 22 and September 22.
These dates also mark the changes of our seasons. Winter turns to Spring on March 22, Spring turns to Summer on June 22, Summer turns to Autumn on September 22 and Autumn turns to Winter on December 22.
However, back in the year zero, these things all happened three days later than they do today.
What happened? Did the earth shift by three days?
What actually happened is that the ancient astronomers were not able to be precise in their calculations. They knew that a year lasts more than 365 days. A year actually lasts about 365 and 1/4 days. Therefore, they added in an extra day every four years, which became known as a leap year.
However, a solar year really lasts slightly less than 365 1/4 days. To correct for this, once every 400 years there is now one year which is not a leap year. That year in which one day is lost is upon us, because although there have been leap years in 1988, 1992 and 1996, the year 2000 would not normally be a leap year. There would be no February 29, 2000.
It took about 1200 years before the astronomers realized that they were or should have been losing one day every 400 years. During that 1200 years, the shortest day of the year shifted from December 25 to December 22.
This error was then corrected, so that from then on, every 400 years there is a year every four years which is not a leap year.
However, because a year is very slightly longer than 365.24 days, one more day has been added back in, so that the year 2000 will be the first year ever which is divisible by 400 and which is also a leap year.
In Rome at the time of the birth of Jesus, there was a traditional celebration on December 25, the shortest day of the year, based upon the Saturnalia, worship to the sun. The Christians adopted this celebration as their own, and celebrated the birth of Christ on that date. Ever though the shortest day of the year later gradually changed from December 25 to December 22, the Christians continued to celebrate the birth of Jesus on December 25.
So, that is the reason why Christians today celebrate the birth of Jesus on December 25.