Why does the romance open at Christmas? How does this relate to the Green Knight's entry?

You might consider the historical context of the story to answer your question. During the medieval period in Europe, which is the setting for the story, the Catholic church was the single greatest entity and the most influential institution in both culture and society. The church's influence extended even to defining the conduct of those individuals (knights) who were tasked with the protection of the king, the land, and the honor of the church. Knights followed a code of chivalry, which included such idealized attributes as honor, courage, respect for women, gallantry, and piety (or adherence to the Christian faith).
Christianity was a tremendously powerful institution during the Middle Ages, and Christmastime—affiliated as it is with the birth of Jesus Christ—was therefore perhaps one of the most esteemed religious holidays. It is possible that Sir Gawain and the Green Knight is set at Christmas because of the sheer importance of the Christian faith in medieval European culture.
In considering the meaning of the Green Knight's entry into King Arthur's hall at Christmas, bearing a "holly twig" in one hand, you might think about the historical symbolism of the holly tree as it related to the Christian faith. For example, the red berries were believed to symbolize the shed blood of Jesus Christ in the crucifixion, and the thorns were believed to symbolize those in the crown of thorns forced onto Christ's head by the soldiers just before he was crucified.
https://treesforlife.org.uk/into-the-forest/trees-plants-animals/trees/holly/holly-mythology-and-folklore/

https://www.ancient.eu/Medieval_Chivalry/

https://www.britannica.com/topic/chivalry