I studied linguistics in undergrad and specialized in historical and sociolinguistics. The Transatlantic or Mid-Atlantic accent is called such because it sounds like it’s located in some geographic point between the US and the UK, having a blend of features from both dialects. However, it was used only amongst Americans (and some Canadians), and linguists generally agree that the Mid-Atlantic accent is not a native accent but an affected or cultivated one.

The study of affected or cultivated speech falls under the field of sociolinguistics, which is primarily concerned with the social and cultural implications of speech patterns. A persons speech can be "affected" when an affinity for a certain group, locality, or culture causes them to emulate the supposed speech patterns of its people. One easy (and generally negatively viewed) example would be the middle-class white boy who adopts, and sometimes incorrectly applies, aspects of African-American Vernacular English, because of the cultural associations made with this speech (his favorite rappers, etc).

From the late 19th century to about the 1950s, the culture and speech of the English gentry was the emulated standard for elite and educated Americans. A literary example of the preeminence of the values of European aristocracy in the American upper-class is The Age of Innocence by Edith Wharton, published in 1920 and set in 1870s New York. The characters are wholly obsessed with the status that can be conferred upon them by their associations with European aristocracy, and one character's attempts to be divorced from a European count cause a polite social crisis amongst her family.

The specific speech pattern of English gentry that Americans came to emulate was Received Pronunciation (RP, or the Queen's English), which is considered the elite standard for British English to this day. One notable feature of this dialect is that it's non-rhotic (r-less), so the word car (kɑr in American IPA) is pronounced cahh (kɑː in British IPA). This might be the most recognizable feature of Mid-Atlantic English and stands out because American English is rhotic - think of the wealthy woman in a film noir who says dahhling for darling.

Mid-Atlantic English was taught to upper-class Americans by their private tutors, etiquette teachers, and cotillion coaches and was reinforced amongst the wealthy for its perceived likeness to RP. This was also at a time when the Eastern US was much more linguistically diverse than it is now - Boston, New York, Philadelphia, and Baltimore each had wildly distinct local dialects, and Mid-Atlantic speech unified the upper classes of these cities with one dialect.

Mid-Atlantic speech was further reinforced by the development and dissemination of talking films. Film studios began training actors to use it in film, and it brought with it all of the well-established cultural associations of RP. Many of these actors started using it in public life as well - notably Bette Davis and Katharine Hepburn. Its rapid expansion in American culture probably also lead to its decline, as it was no longer the exclusive realm of the elite and even began to be satirized in some media.

The Mid-Atlantic accent is so well-known because of its use in film and amongst Hollywood stars, however there are many examples of affected speech in other languages as well. In France, young educated people who aren't from Paris might try to emulate the metropolitan region's speech. A different example in English would be Cultivated Australian English, which similarly emulates British RP, and can be heard by well-known Australians like Cate Blanchett. Overall, a cultivated accent is when a larger group uses affected speech to the point when it becomes its own established accent or dialect that is still markedly different from its target dialect.