According to Johnny Cash's autobiography Cash: The Autobiography, written in 1997 with Patrick Carr, Cash claims that:
...by 1968 [when the Johnny Cash At Folsom Prison album was released], I'd been doing prison concerts for more than a decade, ever since 'Folsom Prison Blues' got the attention of the inmates at the Huntsville, Texas, prison in 1957. They'd been putting on a rodeo every year, and that year the prison officials decided to let them have an entertainer too; they asked for me.
The concert in Huntsville was a success, according to Cash (despite their electrical equipment shorting out in a thunderstorm):
Word got out on the prison grapevine that I was okay, and the next thing I knew, I got a letter from San Quentin [Jail in California], asking me to perform at their annual New Year's show on January 1st, 1958. I went ahead and did that, and did it again for several years in a row.
Mind you, Cash tells a great story (as you'd know from hearing 'A Boy Named Sue') but reality was a little different, according to Robert Hiburn's book Johnny Cash: The Life. Cash was not the first 'entertainer' or even country singer to appear at the Huntsville Texas prisoner; Eddy Arnold had performed at the rodeo in 1951. And Hilburn thinks that instead of the 'prison grapevine', it was Cash seeing the reaction of the prisoners to his music that made him want to do more prison shows, and he tasked his manager Stew Carnall to get more prison bookings, which he did within days. And Cash's first show at San Quentin was actually on January 1st, 1960. After that show, the San Quentin prison warden invited Cash back any time he wanted.
After many years of playing prisons, there was, according to Hilburn, a 1966 plan to record a live show at the Kansas State Reformatory which was cancelled due to Cash's drug addictions leading to unreliability and a lack of ability to stay focused for the recording. In 1967, Cash got a new producer, Bob Johnston, who didn't know about the 1966 plans, and who thought the idea of recording at a prison was great. This resulted in the Johnny Cash At Folsom Prison album (Folsom Prison being in Sacramento, California). From the Folsom Prison side of things, the concert was organised by one 'Coach' Lloyd Kelley, the prison's superintendent of recreation.
This album was a massive success, and so a sequel was planned, inspired by interest by British producers at the Granada TV station for putting together a show about Cash. As a result, Johnny Cash performed at the larger (and tenser) San Quentin (near San Francisco) on February 24, 1969. The Granada TV special is packaged as part of Sony's 2006 three disc Legacy edition of the 1969 Live At San Quentin album; the TV special spends a fair amount of time focusing on the fact that executions took place at San Quentin, in addition to footage of Cash performing.
On that TV special (available piecemeal on YouTube), there's interviews with inmates which portray the inmates as basically average Joes not much different from the rest of the population apart from some stupid decisions; one unnamed San Quentin inmate says something to the effect of 'most of us are really in here for the crime of being poor'. The warden interviewed takes pains to illustrate just how brutal inmate life is. The TV special doesn't explain specifically why the concerts occur, but the implication of the interviews with inmates and the warden is that it's a small mercy to regular people who've been put in a brutal situation.
The view of the prison officials at San Quentin about how to treat their inmates, is explored in a chapter of a 1973 book by the sociologist Erik Olin Wright called The Politics Of Punishment. Wright argues that "in the view of most prison officials [at San Quentin] the two basic functions of prison are custody and rehabilitation" (but that there were a sizeable minority of staff who believed that the role of punishment of prisoners who "had it coming" was more important than rehabilitation). The prison officials believed, according to Wright, that "we are here to teach conformity" and that they hope that "some of these habits will continue on the outside and that the prisoner will learn to conform to the rules of society". Training materials for guards new to working at San Quentin in this era promoted a 'firm but fair' ethos, as opposed to cruel treatment of the inmates.
This official ethos obviously played some role in the prisoners being able to attend annual shows featuring performers like Johnny Cash, and in the fact that there was even a 'superintendent of recreation' at Folsom. Permission to attend such shows could be seen as a reward for conformity to prison rules and good behaviour. After all, being able to experience 'top-grade' entertainment in person would have been a very strong highlight of what was often a brutal and depressing prison experience (as those interviewed by Michael Streissguth for his 2006 Johnny Cash biography attested). Especially in a prison like San Quentin, where there was a prisoners' strike in 1970, and where there had been a guard held at knifepoint by a prisoner the week before Cash's 1969 concert, the privilege of attending Cash's concert (and perhaps further privileges like being allowed to be seated closer to the action on stage) was clearly a strong bargaining chip for prison officials on their mission to 'teach conformity' to the inmates.
The inmates could holler and whoop at Cash during the concert, and Cash could say outrageous things on stage - "if any of the guards are still talking to me, could you get me a glass of water?" - and sing about how San Quentin is a hellish place to be. But ultimately Cash's concert was used as a carrot to get the inmates to conform to behaving in what the prison officials considered socially acceptable ways.