It is correct that, until the late 40s, radio shows were recorded onto transcription discs. These discs are the only reason that we have the shows nowadays. Transcription discs were still used in various ways after tape came along, but tape was the preferred standard once Crosby started using them.
Syndicated shows (that is, shows that were sold to individual stations to play whenever they wanted) were recorded and mailed out to the stations on a week-by-week basis (I think). Shows like Jack Benny, which were broadcast over an entire network, were at one point performed twice in a day...once for the east coast, and once later for the west coast. I think later on they might have recorded the show once and played the disc for the west coast broadcast, but that's something else I'm not sure about.
Sometimes these shows were recorded for various reasons. A sponsor might have wanted a copy of the show for their archives, or the performers themselves might have had them recorded for their own personal use. Sometimes stations would record the show to test equipment as well, and you also have the folks at home who had those little do-it-yourself machines where you could record sounds onto a phonograph record at home. Some people recorded parts of radio shows like this as well (although the quality usually wasn't very good).
The fact is, the reason any of these older radio shows still exist are because somebody along the way recorded it onto a transcription disc, and somehow it withstood the test of time. This leads back to the question I asked a few posts ago. I wonder what the source material is for the older Jack Benny shows. I read that Fred Allen had transcriptions of a lot of his own shows, and after he passed away, they were donated to a library (where, sadly, they still sit today, making it difficult for anyone to be able to hear them without going to the library itself).
Anyway, that's my two cents worth.