How should I connect my dimmer switch?
Those two bottom wires are connected internally in the switch, so if one is hot they both would read hot while connected to the switch. One of the wires daisy chains the hot to other outlets. Both of those wires need to be connected with the hot to the dimmer switch. You may need a larger wire connector than one that might have come with the switch. I know this is a DIY advice webpage, but if those two bottom wires are both hot when disconnected from the switch then it isn't a DIY problem. 

Back some time ago in the previous century it used to be accepted practice to not ground light switches, those days are gone. Some dimmers require a ground to function properly, try folding out the wires in the box to locate bare wires that should be connected together and to the box. If you cant find a ground wire in the box then the ground might be accomplished by a cable assembly or raceway system that grounds the box. If you can identify that one of those systems are used you will need to find a 10-32 green ground screw to put into threaded hole in the back of the box.

Also it looks a little like the glue on the tape on the red wire may be failing, you might want to replace the tape or put a correctly sized wire connector on that before you tuck it back into the box.