The key difference between active and passive voice is that in active voice, the subject of your sentence does something, but in passive voice (using those "being verbs") the real subject is often hidden. The two biggest reasons to avoid passive voice are to make your writing more precise and confident.
Imagine that you're a lawyer preparing for court to defend your position. It isn't going to persuade anyone to say, "Hamlet had been framed by his enemies." You need to spell out exactly what happened with a statement such as, "Claudius plotted to make Hamlet look insane." Take another look at the sentences marked out by your teacher and try to rephrase them to include more specific information about who did what.
Besides being more precise, active verbs also make your writing sound more confident (which is obviously a benefit in a persuasive essay). Compare these two examples:
Passive: A speech was given by the delegate from Michigan, and a challenge was issued by him to everyone attending.
Active: The delegate from Michigan gave a speech and issued a challenge to everyone attending.
Passive voice has a way of sounding uncertain, like maybe you're trying to hide something or maybe you're not really sure about what you're saying. Be bold and use those active verbs to really put your thoughts out there!