off

Ah, here. This one (among many, many others):

go off […] To start into sudden action; to break into a fit of laughter, extravagance of language, irrelevant or unintelligible discourse, etc. (emphasis mine)

Ohhhhh. Now I'm stumped as to how to explain/justify “go off” in that sense. But I have had a wonderful time considering all possible meanings of off.

A contranym: it means both off *and* on (e.g., "the alarm went off").

Cricket jargon - the side of the field closest away from batsman's body.