People tow trailers with a class C for the same reason they tow them with any other vehicle-to get them where they need to go. ;) Not sure about your area, but where I live I see Cs towing boats, race car trailers, snowmobiles, everything. By looking at ratings, the larger (mostly) Ford based Cs are limited to 3500 or 5000 pounds towing capacity. But lots of people ignore that, I've seen several Cs towing large tandem axle enclosed trailers without issues. I sometimes towed a smaller 6x12 enclosed with mine, and the only time I knew it was there was when I was turning a corner and could see it in the mirrors.
And yes, the "gizmo" on your motorhome's dash is for activating the electric brakes on a trailer.