Although the skit is primarily focused on Politics, Barry will always lash out at the guests and give them idiotic threats. the lashing out is refrencing a press interview which a reporter asked who he was, causing him to yell "I'm Barry effin' Gibb!" which is Barry's catchphrase.
Robin is normally silent throughout the skit, but sings along in the theme( which is a parody of 'Nights on Broadway') and when Barry asks him to introduce a guest. Barry will always beg his brother to talk, Robin responding with a quiet "I don't want to."
You can normally find recorded scenes of the 'Barry Gibb Talk Show' on Youtube.