It's the world we live in now, you are talking of a world where their no video games, 1000 channels, and no internet.
The entire nation was captivated. This was way before internet and unless you were physically at the game, listening on the radio or watching the game on TV, there was no way to find out how he did until the 11PM local news. Most of the games at the time were only available on TV locally, the only national broadcasts was "Monday Night Baseball" (with Howard Cosell) or some other network "Game of the Week" on a Saturday or Sunday afternoon.
The news wires however (like AP or UPI) would send the results of each Pete Rose at bat on the wires but only the media had access to those and it involved having some sort of teletype machine on hand to print the wires out.
So a lot of radio music stations would announce the results of a Pete Rose at bat between songs.
For example, "Just What I Needed" by the Cars would finish playing the the DJ would exclaim "PETE ROSE JUST SINGLED IN THE 4TH INNING AND HIS HITTING STREAK IS ALIVE AT 38 GAMES". Then the DJ would launch into "Sultans Of Swing" by Dire Straits or some other song that would tie in somehow to Pete Rose or baseball in general.
"Boys Of Summer" by Don Henley would not get released until some years later - I think that was 1984.