First, it’s important to know that there are other colours associated with Christmas, liturgical colours are white and gold, green and silver are also associated with Christmas.
The red colour in Christmas represents several things and that is why it is common.
The red colour was the part of the main prop used in a popular medieval play about Adam and Eve. It represented the apples on the tree in the garden (paradise tree).
The red is also the colour of robes that Bishops wear. The closest relation to this is that red robes Santa Claus (St. Nicholas) wears.
To many others, the red colour represents the blood shed by Jesus Christ during his crucifixion.