Firstly, I think you will find an anecdote much easier if you understand what exactly that is. An anecdote is usually an interesting short story, sometimes humorous, about a person or incident. It is used to engage readers as well as illustrate a point. For example, take a look at the passage below from Emerson’s essay:

I remember an answer which when quite young I was prompted to make to a valued adviser, who was wont to importune me with the dear old doctrines of the church. On my saying, What have I to do with the sacredness of traditions, if I live wholly from within? my friend suggested, — "But these impulses may be from below, not from above." I replied, "They do not seem to me to be such; but if I am the Devil's child, I will live then from the Devil." (“Self-Reliance”, Ralph Waldo Emerson)

One of the key phrases here is “I remember” which draws your attention to Emerson telling a story about himself as a young boy. Emerson uses some humor in stating that he is the “Devil’s child.” Essentially, he is using this anecdote to further the purpose of his essay; that is, to rely on your self as the title suggests. Emerson is attempting to explain, by using an anecdote, that even if he were the Devil’s child, he would still embrace his identity and opinions as an individual while encouraging his readers to do the same. Nothing could ever convince him to change or rely on someone, or something, other than himself. Anecdotes usually work as a very effective method of persuasion.