There were a number of famous musicians who played notable concerts in prisons during the 1960s and 1970s, oftentimes recording these shows for albums and/or film. Perhaps the most famous instance was Johnny Cash at Folsom Prison, but there were a number of other such performances - BB King's Thanksgiving concert at Sing Sing; Frank Sinatra at San Quentin, the Grateful Dead at Terminal Island.
What prompted this scene to arise and why did it languish?
In a previous answer, I discussed the reasons why Johnny Cash played at Folsom and San Quentin, going into differences between Cash's autobiography and a biography, and into sociological studies of the San Quentin prison. As to why this scene seems to have stopped, there was some sympathy for 'outlaws' in the 1960s and early 1970s in popular culture in general; you get sympathetic portrayals of criminals in movies like the 1967 Bonnie & Clyde. However, from 1960 to 1990 there was a dramatic increase in crime rates; the UCR Data Tool claims there were 288,460 reports of violent crime in 1960, and 1,932,274 reports in 1992. In an environment of increased crime - and accordingly where politicians were increasingly likely to run on 'law and order' platforms - sympathy for regular prisoners became politically less viable both for politicians and musicians. So, if this - very long - list of Johnny Cash live performances is accurate, Cash continued to play prison concerts until about 1974-1975, and then stopped, apart from a couple of prison performances around 1980 (one in Australia).