There's a more detailed account of it in Steve Goodman's biography. Goodman was opening for Kristofferson, and Kris was blown away by Goodman's performance. Steve was doing some of his own songs (obviously by this point he was doing "City of New Orleans" because he had recorded that for an Earl of Old Town LP in 1970) and "Donald and Lydia" (which ended up on his first album, which Kristofferson produced) and "Sam Stone" from Prine's catalog. Kristofferson suggested that Paul Anka, who was involved in Buddha Records, sign Goodman. (The famous quote was after Anka asked Goodman if he'd like to go to New York to record Steve replied, "Would you like to see a short, fat Jewish kid dive into a bowl of chicken soup?") Steve, however, the ever-generous person that he was, said, "Listen, if you think I'm good, you need to hear my friend, John Prine."
So Kristofferson, Anka, and Goodman went to the club that Prine had been playing at. It was a Sunday night, as Prine remembered, and all the chairs were on the tables and John had just been waiting around to be paid when Goodman called him and told him not to go anywhere. The entourage arrived and they had John play six songs -- then play them again. Kris then said to Anka, "I guess you're going to buy TWO plane tickets to New York."
There's a DVD of Goodman's, "Live From Austin City Limits and More!", that has an extensive interview with both Kristofferson and Prine about that night.
BTW, I think the liner notes you read are on "John Prine," not "Sweet Revenge."
Best album about divorce: Prine's "Bruised Orange."