I don't know specifics for the 2009, but it should bolt to pre-drilled holes somewhere on the subframe or a reinforced area of the unibody. The hitch should come with instructions unless you bought it used.
That electrical connector does sound like the trailer wiring harness. You can check for 12 V when you have your tail lights on.
Make sure the bolts are "nice and tight". If the mounting points are sheet metal against sheet metal, you can look up maximum torque values for the size and grade of bolt being used. If it mounts to someting "crushable", the instructions might give you torque values. The idea is that the clamping force generates enough friction to keep the hitch in place, as bolts are weaker in shear than in tension. However, clamping force will be reduced if bolts are over-tightened into the plastic deformation range (and the bolts will likely break soon after).