This is a phrase which people use to transition between two ideas in a conversation, two parts of a speech, two sections of a meeting, etc. You're saying "let's move on to the next thing, which is related to what was just said.
For example, imagine you're leading a parents' meeting at your child's school, and you've been talking about raising money for the school band. Then one of the parents started to complain about the food in the cafeteria. You were going to discuss requests for improvements next anyway, so you say this:
On that note, let's move on to new requests. Does anyone have any other areas that they'd like to see improved?
The word "note" in this expression means "topic", "theme", or "emotion".
You can also use "On that note..." ironically to end a conversation. Use it when the conversation has started to break in some way, like when someone becomes too silly, complains about something that they're not supposed to (like your boss), or gets upset. You can use it like this without even finishing your sentence:
A: I don't care what any of you think! I'm going to do whatever the hell I want!
B: Well, on that note...
When you use "On that note..." in this way, you continue to appear calm and civilized by not responding directly to the inappropriate situation, while also pointing out that the conversation has gotten a little out of control.