What do we mean when we say it today?
The Queen's (or King's, depending on who's the head honcho at the time) English is a phrase used to mean the English that people in the United Kingdom speak. Have you ever noticed how some Brits say flat instead of apartment, call a trunk a boot, and ask for a biscuit in the place of a cookie?
That's because even though Americans and Brits both speak English, they have slightly different words for some things. If someone refers to "the Queen's English," that person is talking about all those words the peeps across the pond say. Plus, it's usually said with a hint of ownership about it.
Some Brits are all "We speak the proper way over here, unlike you Yanks." Usually it's just a joke when this phrase is busted out. But it's still got a twinge of superiority about it. Just like Mistress Quickly thinks she's better than Caius because she speaks proper English, so do a lot of Brits today.