There are two basic types of key switches that include rubber sheets like that.
In the more common type, the actual switch is formed between two mylar flexible printed circuits with a spacer sheet between them. This is the most common "membrane keyboard" design. The spacer has holes cut in each key position, so when a key is pressed the two circuits can make contact through the hole. The rubber membrane is just a protective cover that has domes molded in it to provide a semi-tactile response.
The other type actually uses the rubber domes as part of the switches, by impregnating them with carbon particles. When the domes are pushed in they make contact across two () shaped circuit pads, on either mylar FPC, or regular FR4 PCB. That type is more common on remote controls and other low-duty cycle devices and is much less repairable. Kits exist to replenish the conductive coating on the rubber, but that is only a temporary fix.