how does using an ancedote persuade audiences?

Answer 1

Sometimes, an idea doesn't get explained easily without an example. An anecdote is an example taken from life that shows how a concept works when people apply it.

So an anecdote persuades the audience through showing an experience they might have themselves.

Answer 2

Usually an anecdote is humorous, and humor relaxes an audience and allows them to feel comfortable with the speaker.
If it is to solidify a point, then an anecdote can help the listener relate the point or position and apply it to themselves.