They go on the outside, using a clear adhesive to hold them in place. There are products specifically for this, including the Seaview brand background mounting solution that Petco sells. My guess is that the residue on your tank is from the tape. "Goo Gone" or similar products should remove the residue if it bothers you. As for the colors running - seems to me it shouldn't do that, since it is intended for aquariums, but then that would make sense, so it's probably not done. LOL. Anyway, if it bled outside the tank, imagine how much worse it might be INSIDE the tank submerged in water.

There are mounting solutions made specifically for this purpose. Petco sells one, Seaview brand. You could probably find other brands at other pet stores, and other adhesives sold in hardware stores or grocery stores would probably also work. I would think the most important factors would be that the adhesive is strong enough to hold the weight of the background (probably not much weight), and that the adhesive be clear after it has been applied, so as not to interfere with viewing the background.