Because on most woodwinds including the clarinet, saxophone, and flute there is by necessity a series of "slave" keys that close and open along with whatever tonehole the finger actuates or covers. This as I understand it was an innovation of the Boehm fingering system developed during the mid-nineteenth century intended to improve the acoustic properties of instruments while also allowing for optimal finger position.
The keys immediately under the fingers on each hand are often referred to as the "stacks" and the mechanisms included are quite complex, including one or more slave keys included that close when certain keys are pressed. The Saxophone (most of them, anyways) has one such slave above each stack for the right and left hand. You press any key of either stack down except the G key (left hand ring finger) and the slave mounted above each set closes with it and likewise opens when released. The Clarinet has three slaves, total and the rings are necessary to close and open these. The ring design allows this to be done while still allowing that hole to fully vent. Rare 'plateau' model soprano clarinets lack rings and instead have a 'cup' like a saxophone would. Larger clarinets like the Bass use cups instead of rings.