In "Sir Gawain and the Green Knight," the term girdle here is like a belt: it is something worn at the waist. It is not what has been used by women over the years to pull in one's stomach or "girth."
In realistic terms, I assume your question is less about socks or gloves, and more about why the magical item is a "girdle" specifically, rather than something else.
My guess here follows two paths of reasoning. (And I believe we must guess as we do not know the actual author of the tale or his intent.) First, the girdle was worn at the waist, a place where a weapon such as a sword or knife would be placed, perhaps symbolizing the girdle's significance as a weapon of sorts for protection. The second reason would be, I assume, that it could easily be concealed from view.
Because Gawain does not know that Berilak is the Green Knight, he would keep the girdle secret from his host only because he had agreed to share with Berilak anything he had received during the day. This he had honorably done, until it came to the girdle. (He hides this from Berilak simply because he does not want to die when he faces the giant in battle, and hopes the girdle will save his life.)
The Green Knight is, Gawain assumes, a creature of great strength and magic (remember he puts his head back on), not the same "person" as his host. For the sake of battling the Green Knight, perhaps Gawain does not want his foe to realize that he has had his own help in the form of the magical girdle from Berilak's wife.