Mainly they're traditions adopted from various other celebrations and conglomerated into one. As I mentioned, it's got a very complicated history. But to give you a very short and overly simplified version-
Trees have been used to celebrate winter festivals for thousands of years. Depending on the celebration they can symbolize anything from the coming of spring to everlasting life with God. And the lights are because before the invention of electric lights, the trees used to be decorated with candles.
The gift-giving was adopted by the Christian religion from the Pagans. The Pagans would give one another gifts during this time of year to lift one another's spirits and celebrate the God of Agriculture.