It was exactly like toothpaste, except without the mint smell.

After you polish for a while, you buff it out until it shines.
Then you open a sealed pad that has a damp wipe on it, and wipe the lights. Once that dries, you open a second pad that contain a kind of varnish, and spread that as evenly as you can over the lights.
And then you are done. The pictures don't do it justice. They look remarkable better. These 13 year old headlights.