The following light was removed from my outside front porch because the bulb had burned out. Only problem is I didn't remove it and now I need to put the thing back up, but I'm not sure how the connection should be. Is this something I should even be doing at this point?
It looks like you've ruined it. The stranded wires (such as the black one visible in the picture) are usually permanently attached to the lamp (Inside those cloth tubes). The solid wires are your household wiring. Normally you'd attach the stranded black to the solid black with a wire nut, and the same with the whites. Then attach the bare copper ground (or green) to the brass tab in the light. But it looks like someone just ripped it off and detached the stranded wires from the fixture.
Buy a new one, and if you are unsure of how to remove the old wires or correctly wire a new fixture, the safest thing to do is hire a qualified electrician.
Also, make sure your housemates read this: http://www.wikihow.com/Change-a-Light-Bulb
Warning, dangerous advice, see comments
If the black and red cable is still long enough. Fiddle it through the hole and attach it directly to the bulb's socket.