I could have sworn I’ve done a post on this one, but apparently not, so here goes. The OED‘s definition and first three citations:
chattering classes n. (occasionally also in sing. chattering class) freq. derogatory members of the educated metropolitan middle class, esp. those in academic, artistic, or media circles, considered as a social group freely given to the articulate, self-assured expression of (esp. liberal) opinions about society, culture, and current events.
1980 F. Johnson in Now! 21 Mar. 48/1 The peculiar need for something to be frightened about only seems to affect those of us who are part of the chattering classes.1990 R. Crichfield Among British vii. 457 The old Britain of Eton, Oxbridge, the land, and the Guards, allied with a chattering class of literary intellectuals, so invaluable when it came to running an empire, is deadly when it comes to bringing the country into the 1990s.1994 Daily Mail 18 July 8/2 A battle between Middle England—the sensible heart of the British middle classes—and Islington Person, the politically correct voice of the chattering classes.
Tooling around Google, I found a use that antedates the OED‘s first cite by more than a century, in an 1871 article in The Spectator called “The New Indian Danger”: “… and everything seemed to grow dear at once, to the immense disgust of the chattering classes who bought …” It’s clearly an outlier–nobody picked up on it–but it’s there.
In any case, the phrase is definitely of British origin. This Google Ngram Viewer chart shows it picking up popularity in the US starting in the 1990s, though still lagging far behind Britain.
Ngram Viewer only offers reliable data through 2000, but “chattering class/es” has definitely picked up steam in the U.S. since then. In the New York Times, one or the other variants phrase have been used 295 times post-2000, most recently in a review of a book on Hillary Clinton’s campaign that appeared April 24 of this year: “… while the chattering class may be intrigued by, for example, Clinton’s flirtation with ABC’s David Muir, ordinary readers may find themselves swimming in references to journalists and staffers who are far from household names.”
notoneoffbritishisms.com 
Guards – Eton: I don’t think they are particularly linked with the chattering classes. BBC, Guardian, north London, metropolitan liberals …. that is where to look.
The Chatterati is another version of the phrase.