Dripping toast gives up the second slice on purpose and stakes everything on what the fat does as it cools. Beef dripping is the rendered fat left after a roast, and the part that matters is not only the soft white fat but the dark brown jelly that settles beneath it, the concentrated meat juices set into a savoury layer. Spread thick onto hot toast, the fat melts and soaks in while the brown jelly stays as discrete salty pockets, so a single piece of bread carries the entire flavour of a roast without any meat present. It is an open-face sandwich whose whole argument is that the by-product of the joint is worth eating in its own right.
The craft is timing and the cut of the dripping. The toast has to be hot enough that the white fat liquefies on contact and carries into the crumb, because cold dripping on cold toast sits as a greasy slab and reads as nothing. The bread is taken further than for a closed sandwich, firm and dry, so it holds a heavy fat load without going soft before it is eaten, which is the same reason it is left open: a top slice would only trap steam and collapse the base. A scrape of the brown jelly is dug out deliberately and spread alongside the pale fat, since that dark layer is where the salt and the roast flavour actually live, and a final pinch of salt over the top is the only seasoning the thing asks for.
The variations are a question of how much of the jelly is wanted and how the salt is handled. The pale, fat-forward version is soft and mild and close to plain bread and butter; the jelly-heavy build is darker, saltier, and tastes most directly of the roast it came from; the white-pepper and the toasted-edge traditions each push the same idea a little differently. Those deserve their own articles rather than being crowded in here.