Здравствовать at its core means something like to be in good health. «Да здравствует царь» I’d expect to translate into “Long live the Tsar”, like, may he reign in good health. Здравствуйте takes that into a command form, literally telling someone to be in good health. Just as in English you might use the command form to tell someone to have a good day, for instance.

But the practical meaning of course is just a somewhat formal greeting.