The Greek theater
For Nietzsche, the theater of ancient Greece is a bedrock for all Western art. He believes that although modern philosophy debates about meaning and nihilism, the Greeks pretty much figured it out. He views art as a resistance to the natural nihilism of the human experience, and of those, the most tension-producing art is that of tragedy because (Nietzsche alludes to Aristotle here) tragedy evokes existential pity and dread. The dread is the dread of tragic downfall, which is held in tension with the idea that perhaps life is meaningless.
Apollos and Dionysus
Two modes of human life arise as mainstays in Greek culture and beyond, those of Apollonian life and Dionysian life. Essentially the difference is a split between responsibility to social expectation and a service to one's own animal desires. That is an internal-external split, so perhaps the easiest way of understanding what Nietzsche means by these is to view them through the lens of loyalty. A Dionysian is loyal to his bodily instincts which guide him through desire and pleasure, and an Apollonian fellow would be loyal to his role in the community, accepting responsibility by duty and honor.
Tragedy as meaning
Tragedy is a symbol for human experience and the quest for meaning in life because tragedy involves a transformation of the soul through the catastrophes of fate and karma. Karma is a more Eastern way of analyzing life's tragic quality and focuses less more on positivity, but Nietzsche's analysis focuses mostly on negative experience like tragedy and suffering. Nietzsche's point of view is that tragedy stretches the human audience to a breaking point which once arrived at (climax) promises a reward for suffering in catharsis. That reward system is a mechanism through which humans can find meaning, even if it comes through art instead of daily life.
Catharsis
This word comes through Nietzsche's vocabulary from Greek philosophy. The word in Greek meant "a cleansing" which is a fairly apt description of the phenomenon. The drama and tension produced by human activity and fate is a musical tension which can be resolved in finale. The final chords of a symphony are the catharsis of the piece because they fulfill the emotional expectations which are created through the tension of music. Nietzsche applies this to life as an approach to art theory and humanism.
Ecstasy as an allegorical journey
Touching on a specific style of Greek tragedy found most often in Dionysian stories, Nietzsche describes the allegorical journey from Dionysus to Apollos. The dynamic there might seem counter-intuitive, but it isn't. In Euripides' The Bacchants, we see what Nietzsche means as the public moves through a dynamic which starts in full indulgence. That indulgence in desire and pleasure leads to experiences of divinity that are religious and valuable, but they also lead to tragic fates and suffering. As suffering modulates their experience, they gradually discover the virtues of balance and responsibility.
