The most basic way to analyze a persuasive speech is to ask whether it persuaded you. If, at the end of the speech, you are persuaded, then the speech was persuasive. That said, you should also get the opinion of others, as others may have a different view. The more people are persuaded, the better the speech is.
Second, you should check the facts of the speech. Take a close look at the evidence and see if the evidence is accurate and whether the evidence supports the thesis of the essay. Often you may find that people slant the evidence to fit their points of view. Sometimes people do even more than that and fabricate the evidence.
Third, see if the speech also addresses alternate views. Considering potential arguments is also a mark of a good essay, as it anticipates criticism.
Finally, look at the form of the speech. Content, of course, matters most, but the form of a speech is also important. Does the speech flow? Does it end on a strong note?
But what about language? ( simple, complex, presence of figures of speech). For example, my homework is to analyze Martin Luther King’s speech "I have a dream".
I think that the best way to help someone understand something is to use basic speech (not breaking every word down into the smallest parts, but just common knowledge). To further help someone understand, use simple comparisons.
Like if you were describing how the sun sets, you might add a simple comparison like, "when you look straight forward and turn, your view changes, as does your view of the sky when the earth rotates. It appears the sun is moving, but really, your just moving your sight path".
You might not want to get that basic or else the reader might feel a little stupid for not getting the concept in the first place.