It's a little hard to the connections on the switch, in the box I can see bare ground wires, so it doesn't appear your third wire is a ground, it looks like maybe it is poked into a hole near a screw terminal that would be an alternate connection point connected to a screw terminal. Looks like you have 3 sets of cables, One hot pair coming in, a hot pair continuing to other outlets, and a pair that is switched to the light.
The two wires on one side of the switch both need to be connected to the hot (gold screw/black wire) on a dimmer, the single wire on the other side is the load wire (silver screw/red wire). If the dimmer you are trying to use needs a neutral the white in the box may be a neutral, would probably need a meter to be sure.
If you are unable to attach two wires to a hot screw then you may need to pigtail a short 4" piece of wire and a yellow or tan wire connector to join 3 wires together to attach one to the screw.
Also note older switches often did not have a connection to the ground (bare) wire, your new switch will likely have a green screw or wire that needs to be connected to the bare wire in the box.