The two Greeks
Nietzsche offers two seemingly opposite versions of an ancient Greek person. On the one hand, there were the materialistic Apollonians who believed that life well-lived in the body pretty much meant life well-lived in the heavens. Their lives became about physical and scientific progress, whereas the Dionysians, focused on passion and mystery, developed a new form of art—the highest form of art: The Greek tragedy. In Nietzsche's view, the more literalistic Apollonians were like a ballast that helped solidify the birth of Tragedy as a genre.
The irony of chaos
The irony of chaos in the Greek tragedy is that it shows up when the hero has thoroughly earned it, almost as a karmic punishment for whatever evil pride the hero has demonstrated. The irony is that there is an order to the way chaos functions in a tragedy. Ultimately, the catastrophe is so horrific that the hero's life is permanent disfigured somehow.
The irony of audience
A person reading a book has a private experience with art. A person at live theater who is viewing the play for the first time, that audience member is having a public experience. So if they cry, for instance, everyone sees it. And many of these plays were performed for literal Bacchants, so they were likely very drunk or high or whatever they could manage. Then they were subjected to brutal, terrifying plays about how the gods never let anyone win. The effect is that the audience is forced to experience a deeper sense of awe and reverence for the unknown, chaotic properties of the universe. Dionysians did that for religious purposes, but Nietzsche does it as a science.
The greatest form of art
Nietzsche is a cynic, so for him to offer praise to anything is quite a feat. Yet, there's no two ways about it, Nietzsche is a bit of a theater geek. He views live tragedy as the highest form of art, especially in the ecstatic, bewildering way the Greeks used to do it. The effect was religious in nature, like cosmic horror. Ironically, Nietzsche loves it. He says it's one of the only sufficient responses he has ever found to the problem of nihilism.
Catharsis
Just as pleasure and pain are woven together during sex or drug abuse (like Dionysian binge drinking, for instance), tragedy weaves pleasure and pain together, alongside honor and shame, to create a mechanism for catharsis. Catharsis is the feeling one might enjoy after weeping until they feel finished. The emotional clarity is what Aristotle believed kept people coming back for their tragedy again and again. Nietzsche's exploration of catharsis has more to do with the group psychology of the audience, which is another ironic twist (because catharsis feels personal, and it's weird to imagine "shared catharsis." In this way, Nietzsche views theater as an erotic experience.
