When looking at reasons for Sir Gawain's courage, knowing something about history and the Medieval Period, the period in which Sir Gawain and the Green Knight was written, will help in identifying examples. One of the main themes and characteristics in Medieval Literature and in Sir Gawain is that of the hero. There are different types of heroes, and they have varying characteristics: a person of either common or noble birth, who accomplishes extraordinary feats and possesses extraordinary strength (it would make sense that one would also need, or have, an extraordinary amount of courage in order to accomplish such feats), is mortal (so a hero must be conscious of their own mortality, which also means that courage is necessary in order to risk ones life, regardless of brute strength or luck, etc.), is flawed, must fight for and earn their honor (a noble birth does not automatically grant a person honor), overcome some personal, inner conflict, plus several more, but these characteristics are important ones to know, and will help in identifying examples that will highlight Sir Gawain's courage.
Early on, Sir Gawain shows courage when he stands up to accept the Green Knight's challenge despite his own fear of death. His reasoning for this was to take the place of his uncle, King Arthur who was originally the chosen participant after none of the other present knights volunteered. Sir Gawain couldn't allow his uncle, the legendary king to risk his life, when that of his own was of so little importance.
After accepting the Knight's challenge and beheading the Knight, Sir Gawain must face his fear of death by seeking out the Green Knight the following year in order to have the same done to him (beheading), and knowing his fate, he still finds courage within himself to honor the Knight's challenge. Despite dying, he puts his word, honor, and loyalty above his own life--highly regarded traits in the eyes of knights, and as such, he doesn't want to disappoint his uncle.
All in all, the main reasons Sir Gawain stood up to meet the Knight's challenge in the face of his own fears was to live up to the code of honor that the knights and Arthur highly regarded--Sir Gawain himself held himself to an even higher, stricter code for himself than the other's did and courage showed itself through these actions.