Does "get out" mean here "end" or "finish"?
Yes, that's correct! "Gets out" or "lets out" are casual expressions often used in association with classes or school activities ending. Thus, we can understand "when his harmonica lesson gets out" to mean "when his harmonica class ends." Ex: I get out of school at 3 PM. (School ends at 3 PM) Ex: My math class lets out in an hour.
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