Meaning of "On that note" and how do you use it?

Your assumption about it's meaning is largely correct. It typically is used to transition from one subject to another subject on the same topic.

It's meaning can also be expanded to include associated emotions or feelings. This can be seen in similar expressions like: "On a happier note" or "On a sad note."

It's a metaphor from music that typically comes at the end of a speech and means in that way or in that spirit. The speaker will say something amusing, profound or sombre and then say And on that note, ladies and gentlemen, I would like to end my address to you tonight.