Not entirely sure if this is appropriate, if it isn't I'll delete it, but for those wondering what the transatlantic accent is this article has clips from Hollywood movies using the accent.
The article also answers the OP's question but its not really a reliable source so just ignore that part.
In the late 1800s and early 1900s, the discipline of elocution went through a bit of a renaissance, and was considered part of a young gentleman or lady's proper education. What had been a branch of rhetoric was being taught in (American) public schools, and presumably, any professional actor and speaker would have done well in the subject or taken private lessons. There were schoolbooks like McGuffey's Eclectic Readers with poems for recitation alongside the familiar readin' and writin' lessons. There was one right way to pronounce things, and many wrong ways.
The discipline is lampooned gently in Alice in Wonderland (the poems Alice can't quite recite correctly are part of elocution exercises) and a little more pointedly in later things like My Fair Lady and this scene in Singing in the Rain.
Why it declined? Well, in part I think (I think) it became thought of as stuffy and old-fashioned and kind of a hassle, like we think of business suits today... and it part because with the rise of Modernism, Western culture became less interested in being monolithic and more interested in littler component subcultures. Different accents and ways of doing things were, like, OK, man, and not aberrations or deviations from a more-perfect norm.
I'd bet that in Hollywood, the decline of elocution-style delivery and the transatlantic accent maps perfectly onto the rise of The Method and naturalistic acting... with Marlon Brando in Guys and Dolls (with the cartoon accent and On the Waterfront (with the not-so-cartoon accent) as the signposts marking the end of the era.
Two remarks:
a) wasn't it rather called the Mid-Atlantic accent"?
Because the meaning of "transatlantic" changes, depending in which country you're using it: in the US, it refers to Great Britain, and vice versa. https://en.wikipedia.org/wiki/Mid-Atlantic_English
b) It didn't decline entirely, but still can be used by actors to create and place their characters, the two brothers Crane from "Frasier" being perhaps the best-known examples.
hi! you may find this section of the FAQ of interest:
Two notes:
For curious readers, a well-known example that you may be able to hear in your head without recourse to a sound sample is Jackie Kennedy (Onassis); if you do want a sample, her White House tour is easily found on YouTube.
This subject has been discussed in some detail in /r/linguistics; it's worth a search there.