What does it mean for an old time radio show to be "Transcribed"?

"Transcribed" means the entire show is/was written down as recorded and broadcast.

To complicate things, in British English, a transcription would probably be a written record of what was said in the show. In America, During the golden age of radio and in the age of heavy censorship, extensive records were required by the government.

In addition they needed to be able to demonstrate to advertisers that their copy was read/delivered as paid.

For example Johnny Dollar uses the phrasing--"The transcribed adventures of the insurance investigator with the action-packed expense account."