These Telephone Jukebox systems ran in a number of US cities for a while in the mid-twentieth century including Manhattan (setting in the film). Music was transmitted over premium quality phone lines to the jukebox and while not what we would consider high quality these days was probably sufficient bandwidth for a 78rpm record.
Rock-Ola was the producer of such a device, the rather fine looking Mystic Music Jukebox. The central office served up to thirty jukeboxes. The Rock-Ola boxes also had up to twenty records stored inside it locally so users wouldn't always have to use the remote service (the machine in the movie seems to have a list on the front of it, maybe these were the records stored on it). The key advantage of a telephone jukebox being, of course, the much wider range of music that was available, from a few dozen held locally to a few hundred .
One of the most successful of these devices was the Multiphone by Kenneth C. Shyvers which at one stage had eight thousand machines in use. This system required two dedicated phone lines, one to speak to the operator (based in Seattle) and the other for the music . These systems began to appear in the US in the late 1930s and survived in cities like Washington until the late 1950s. By then the improved technology of 45rpm singles (which of course had better sound quality than 78s and also were more compact) rendered the Telephone Jukebox (which wasn't a cheap system to run) obsolete.
|Scenes from the movie|
|The operators look for a record|
|A considerable collection of 78s|
 Barry Ulavov, "The Jukes take over swing", American Mercury (October 1940) p. 177
 Gert J. Almind, Coin-Op Telephone Music <http://juke-box.dk/gert-telephone-music.htm>