Light switch with 9 wires
This is not overly complicated and is quite typical.

The part of the old switch with two black wires: Remove both wires, splice them with a wire nut and one of the black wires from the dimmer.

The other single wire on the switch: Splice with the other black wire from the dimmer.

The green form the dimmer goes to the ground(s). 

LEAVE EVERYTHING ELSE ALONE.