What do we mean when we say it today?
Most people shorten this to "the play's the thing" nowadays, and bust it out when they're talking about, well, a play. Theaters and acting companies like to use Shakespeare to do their advertising for them, and if we are talking about plays, it's a fun way of drawing attention to whatever is being performed.
Now you see why people can't have the last part of the line: "wherein I'll catch the conscience of the king" because it just wouldn't make sense. Unless you're talking about Hamlet's revenge plot against his uncle, there's no conscience-catching anywhere to be found.