A "humorous aside" is a side note, a deviation from the main theme of a piece of writing that provides an amusing fact or idea. It is not necessary for the main theme, but it provides some entertainment and interesting knowledge for the reader.
An "emotional appeal" is an attempt to get the reader or listener to respond emotionally rather than rationally to the material. Politicians like to do this, they tell stories to get people to think they care and to "convince" them to follow the politician's plan without really examining the plan or whether it will solve the problem.
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