It's part of the Betrothal, not the Crowning, and dates to Roman times. Originally, the Betrothal was a distinct service that could occur quite some time before the Crowning (often years, in the case of arranged marriages among the aristocracy). But as betrothal was considered as binding as Crowning itself, and posed an impediment to sacramental marriage if broken or if one party died before the Crowning, the tendency was to move it up closer to the Crowning, until finally it took place immediately prior to the beginning of the Crowning, usually in the Narthex. In some places, the Betrothal has been moved into the nave itself, and the service merged into the Crowning--though this is an innovation and an abuse (probably from imitation of Latin and Protestant services). However, even in this case, the exchange of rings remains part of the Betrothal Rite, and not the Crowning itself.
