They are just not nice people. Not nice to each other and not nice to third parties. Don Henley wrote notes to chamber maids in hotels how they had folded the tip of the toilet paper roll the wrong way - he wasn't joking, he was genuinely offended. He's that type of person.
That doesn't make them bad musicians or songwriters. Or that Henley can make an observation or two and Frey be a honey-dripping crooner. Bob Dylan isn't always a nice person either.
It's kind of reassuring that The Eagles, behind all the Westcoast piousness and mellow music, are such a dysfunctional and sometimes outright nasty bunch. It would be unbearable if they were actually nice guys and liked each other. Ironically, I played Felder's and Walsh's solo albums for much of the day.