Why December 25 ?

Christmas had also been celebrated by the early Church on January 6th, when they also celebrated the Epiphany (which means the revelation that Jesus was God's son) and the Baptism of Jesus. Now the Epiphany mainly celebrates the visit of the Wise Men to the baby Jesus, but back then it celebrated both things! Jesus' Baptism was originally seen as more important than his birth, as this was when he started his ministry. But soon people wanted a separate day to celebrate his birth.
Most of the world uses the 'Gregorian Calendar' implemented by Pope Gregory XIII in 1582. Before that the 'Roman' or Julian Calendar was used (named after Julius Caesar). The Gregorian calendar is more accurate that the Roman calendar which had too many days in a year! When the switch was made 10 days were lost, so that the day that followed the 4th October 1582 was 15th October 1582. In the UK the change of calendars was made in 1752. The day after 2nd September 1752 was 14th September 1752.
Many Orthodox and Coptic Churches still use the Julian Calendar and so celebrate Christmas on the 7th January (which is when December 25th would have been on the Julian calendar). And the Armenian Apostolic Church celebrates it on the 6th January! In some part of the UK, January 6th is still called 'Old Christmas' as this would have been the day that Christmas would have celebrated on, if the calendar hadn't been changed. Some people didn't want to use the new calendar as they thought it 'cheated' them out of 11 days!
Christians believe that Jesus is the light of the world, so the early Christians thought that this was the right time to celebrate the birth of Jesus. They also took over some of the customs from the Winter Solstice and gave them Christian meanings, like Holly, Mistletoe and even Christmas Carols!
St Augustine was the person who really started Christmas in the UK by introducing Christianity in the 6th century. He came from countries that used the Roman Calendar, so western countries celebrate Christmas on the 25th December. Then people from Britain and Western Europe took Christmas on the 25th December all over the world!