I couldn't access the link, so I'll answer your question based on what you put here. The word 'cathartic' is a synonym for the adjectives 'healing' and 'therapeutic', and the phrase 'teeming with conviction' indicates that it is full of, or alive with, conviction (i.e. strongly held beliefs); therefore the sentance means that the song is both healing to the listener and full of the singer's strongly held beliefs. That said, I personally agree with Johnny's comment that whoever originally wrote that sentence was being pretentious.

I don't think that it means anything. Or maybe it means that the writer likes to use big words to demonstrate that they are really, really smart when they are actually pretentious, condescending, and boring. The good news is that you're getting a good vocabulary lesson out of this!