This incident always fascinated me, having watched it happen in 1997. Back then, I couldn't work out why they were so offended. Watching it now I'm not sure any chat show host, serious or otherwise, would be quite so rude today.
You can see the moment each Bee Gee checks out of the conversation. Robin goes first, early on, when Clive calls them "hit writers, but one letter shy" (at about 3'20''). Barry goes when Clive calls them "tossers" (around 5'45"). Maurice is the most patient, playing along with a smile until the final insult (9'00"), when the brothers follow Barry off the stage.