As in my title question. Do they mean a specific region of the US, something else?
General American is a name that sociolinguists often use for the accent that Americans tend to perceive as "neutral" or non-regional.
General American, like British Received Pronunciation (RP) and most standard language varieties of many other societies, has never been the accent of the entire nation. However, it has become widely spoken in many American films, TV series, national news, commercial ads, and American radio broadcasts.
The General American accent is most closely related to a generalized Midwestern accent and is spoken particularly by many newscasters. It is thought to have evolved from the English spoken by colonials in the Mid-Atlantic states, evolved and moved west. Walter Cronkite is a good example of a broadcaster using this accent. This has led the accent to sometimes be referred to as a "newscaster accent" or "television English".
In addition to Cronkite, a native of Missouri, a number of other early U.S. television newscasters hailed from the Midwest or the Great Plains states, such as John Cameron Swayze (Kansas), Eric Sevareid (North Dakota), and Chet Huntley (Montana). Huntley also attended college in Washington state, as did Spokane native Edward R. Murrow; the standard Pacific Northwest accent closely resembles General American.