The problem of hazy headlights is widespread in newer cars that have their lenses made out of plastic. It has a UV-resistant coating that, in time, gets damaged and starts losing its protective capacities – then the plastic gets burnt by the Sun and voila, you have hazy lights.
There are a few methods to deal with this, ranging from the really easy to the moderately difficult and time consuming. Yes, toothpaste does work, but the effect it gives is limited and it doesn’t offer a longer term solution.
For that you need to buy a kit that works with any electric power drill and some sandpaper. You then grind away all the damaged material from the surface and apply a new UV coating of your choosing.
The most common techniques are explained in the Drive video posted after the virtual jump. However, keep in mind that in some cases buying new lenses (where possible) is probably a better long term solution.