A good place to start with these assignments is to know the definitions of the words, or look them up if you don't. Even if you don't know their definitions—pause here first to guess! I like Kathleen's idea of taking a moment to guess which ones are adjectives, which are verbs, and which are nouns.
If you're unfamiliar with any word here, try reading their entries in the dictionary to find examples of them being used in a sentence. If you need more help, reading from multiple dictionaries (online or hard copy) may help clear up any confusion, because each one will have a different sample sentence.
Now that you know their definitions: What is something that is cathartic? What is something that is caustic? What is something that you could cauterize? What would cause someone to cavil, or what is a cavil someone could have? Answering those questions gives you the basic plot points of your story. Can you answer them in a related way, and weave them together?
(Kathleen gave another very good tip: There may be a way to link "caustic" and "cauterize" closely in your story.)