It dates back to the days when a garter would be part of a lady's normal dress. Certain items of the bride's attirer, including the garter, are thought to be lucky, and the tradition for throwing it started in France, and is the male equivalent of the bride throwing the bouquet. the catcher of either is supposed to be lucky, and some believe that each will be the next to marry. In some ceremonies, it is tradition for the bouquet and garter recipients to dance with each other.
Isn't the garter not also usually the "blue" part of the "something old, something new, something borrowed and something blue" that a bride is supposed to wear?