The story of what happened to the Justice League of America is only hinted at in The Dark Knight Returns. Frank Miller always wanted to tell that story, and perhaps one day he will finish it. But the careful reader can discern most of the details from Superman's soliloquy in The Dark Knight Returns.
From Superman's inner dialogue, we can begin to piece together what happened. As the JLA grew more prominent, a backlash grew in the court of public opinion. First, there were parents' groups - presumably opposed to the 'bad influence' of vigilantes, and the endangerment of child-superheroes like Robin. Then, the JLA was called in for questioning by a Congressional subcommittee.
As pressure mounted, Batman - brash, arrogant, and uncompromisingly idealist - refused to back down. Instead, he laughs. Sure we're criminals. We've always been criminals. Cracks began to form in the leadership structure of the JLA.
Jason Todd died. Batman retired. Things escalated. The government formed a PBI - a Paranormal Bureau of Investigation - to 'hunt down' superhumans. And there was that trouble with Oliver. Green Arrow went rogue, refusing to get with the program. And Superman had to make a choice. Compromise with the public, or go to war with them.
He chose to compromise.
Clark took down Oliver, and retired from public view. And the rest was history. Without Batman, Clark, or Green Arrow - and bad blood between all three - the rest of the Justice League members disbanded quietly. Diana [Wonder Woman] went back to her people. Hal [Green Lantern] went to the stars. Superman continued to operate behind the scenes, as a government agent.
And there you have it - the sad story of what happened to the Justice League of America in The Dark Knight Returns. It is a tale of greatness lost, heroes fallen, and friendship destroyed. The story is told against the backdrop of Superman massacring a Soviet army, and Batman endangering a child while flouting public opinion. Neither man has changed very much, and the two are once again on a collision course.