This practice, as it turned out, was devised as a way to actually physically protect the bride from the wedding guests.
It derives from a tradition in medieval England and France called "fingering the stocking." Guests would actually go into the wedding chamber and check the bride's stockings for signs that the marriage had been consummated. Further, in France, the bride would shudder with terror at the end of the wedding ceremony because guests would actually rush her at the altar to snag a piece of her dress, which was considered a piece of good luck.
A wedding would end with a battered bride sobbing at the altar in a snarl of tattered rags.
Apparently, these practices were so intrusive and invasive that someone, somewhere, decided to pacify the mob by tossing out the garter.