So they get cooked using the heat of the pasta itself (/some of the pasta water). You have to be pretty quick to 1) make sure there's enough heat and 2) keep from making scrambled eggs.
I'm not a cook nor Italian, but I think the egg yolks in here got cooked in what was effectively a double boiler? In carbonara, the yolks are meant to be gently cooked, emulsifying the sauce without fully setting.
I believe the whites aren't used in the recipe, and would need to be used for a different recipe.
What is being used in the video is pork jowl, known as guanciale in Italian (basically cured pork cheek) . It has a high fat content, even compared to pork belly.
Fat is flavor, so no convincing from butchers is required. Also, it's a cheap cut, so if they want to make money off their customers, they'll want to also sell things like Secreto ibérico, or better yet, jamón ibérico de bellota.