The recording you link is camp horror, which is what Vincent Price is best known for anyway. The rolling R's and other affectations in language and manners are part of
the genre expectations of certain sub-genres of horror (
especially camp horror).
Rolling R's are also common in what is known as
Mummerset accent which is used for
comic effect (rolled R's are perceived as funny) or to show that the character is of an
"aspiring class".
Similarly, the "Shakespearean accent" is part of
genre expectations. Shakespeare can be delivered with a modern accent, but that feels inauthentic to some of the audience (not all) because we
expect Shakespearean language to be pronounced a certain way.
This is part of the aesthetic contract we enter into. The language we expect is
"estranged" (made-strange) and that is what seems to lend the plays their dramatic character. Shakespeare writes before the bourgeois theater with its focus on the individual, the private, the personal. His characters are closer to Greek theater than to Ibsen. The plays deal with
fate and destinies of great people. Thus, it must be estranged. One view is that when delivered in modern English, thus losing their estranged character, the plays become "common" (something more suitable for a realist drawing room drama, for example). Of course one can very successfully argue with this view but it's still a valid view.
Where this came from -- you'll have to ask a Shakespearean scholar.
I would guess it comes from the fact that in Shakespearean times English was rhotic (that's why you'll see in the article I link below a reference to the commonly held belief that
AmE is closer to Shakespeare than
BrE). This would go back to the "authenticity" argument.