Tragedy as a response to nihilism
The nature of Greek tragedy means that heroes necessarily encounter the unknown. Their fates are always reckoned with in the context of the play, so the true subject matter of a Greek tragedy is not pride, necessarily, but the meaninglessness of life. The Greeks knew this about themselves and Aristotle wrote extensively about how art was shaping the culture. The idea of a crowd watching a play that made everyone feel something profound and painful within the context of their group identity is a response to nihilism, says Nietzsche, because it helps everyone understand the greater context of their lives.
Tragedy as the highest form of art
Nietzsche also makes a serious case that Greek tragedy is the height of human art. There are many reasons one might make that case, but to Nietzsche, it's just one criterion that sets tragedy apart from all other art: It correctly diagnoses the problem of human life, namely that it always feels like everything makes sense, even when we're digging our own grave. To make sense of this, consider the hubris of the hero who believes he will never encounter calamity. The appropriate punishment for this (according to the tragic form) is that the gods show the hero the limitations of his humanity. That's exactly what Nietzsche is trying to do as a thinker and writer—he wants to disturb those who have a false sense of confidence about life. Therefore, Nietzsche thinks Greek tragedy is the purest art, because it reminds the culture of chaos and suffering.
The role of the audience
In a work of true artistic genius, a title which Nietzsche would likely credit to any of the great Greek tragedians, the point is very clear. The story is a threat against the audience. But, as opposed to experiencing the story as a character in the story (the way life works normally), all the characters are removed to an audience role. The role of this audience is of much interest to Nietzsche. He speculates alongside Aristotle about why individuals enjoy theater, but he departs from the Greek philosopher on his opinion of the audience. Nietzsche feels as though being in a group allows the participants to feel a sense of security that opens their minds. They might have looked away if tragedy occurred on the street, but in the context of theater, the audience is almost forced to endure whatever emotions the artist can evoke.
The consequences of knowledge
Dionysian knowledge constitutes insight into the innermost realities of our world, and at times The Birth of Tragedy demonstrates that often we are better off not knowing the truth about our existence, universe, and cosmos.
Though The Birth of Tragedy is a multifaceted book with many topics, one of its concerns is the value and consequence of knowledge. And this was a pertinent topic in the late nineteenth-century as traditional European epistemology began to show weakness in the face of scientific advancement. Western culture had relied on the comfortable ideological center of Christianity for ages, yet as modernism progressed rapidly, scientific methods proliferated alongside discoveries in the physical sciences that threatened to explain our world and universe in physical, rather than metaphysical, terms. Darwin's theory of evolution galvanized this already mounting conflict between religious ideology and scientific fact, and nineteenth-century man was forced to re-conceive of himself and his place, purpose, and value in the world he inhabited.
Yet this was a self-inflicted wound, the result of our Socratic impulse for knowledge that, for Nietzsche, accelerated scientific positivism -- which, ultimately, de-centered Christianity. And when this occurred, Western thinkers struggled through existential crises. The collapse of religion in the nineteenth-century caused a widespread psychological and epistemological crisis that Nietzsche worried would result in nihilism: the belief that life is not worth living.