The way that catharsis is related to Romeo and Juliet is that the moment of catharsis in the play occurs when Romeo and Juliet kill themselves.
Catharsis is emotional discharge after something tragic has happened. The audience usually cries and feels pity after something sad has happened, and that moment is called catharsis. So naturally, when these two young people kill themselves, it is quite sad and the audience feels strong emotions after their death.
