In order to examine Aristotle's ideology on tragedy and the audience, one should begin with his definition of tragedy.
“A tragedy, then, is the imitation of an action that is serious and also, as having magnitude, complete in itself; in language with pleasurable accessories, each kind brought in separately in the parts of the work; in a dramatic, not in a narrative form; with incidents arousing pity and fear, wherewith to accomplish its catharsis of such emotions.”
Let us break this definition down.
"A tragedy is the imitation of an action that is serious and also, as having magnitude, complete in itself"
A tragedy cannot be about a bad haircut, a bruised ego, or a failed test. It must be about something "serious" and important. Given that all tragedies deal with death, it (death) is both serious and important.
"In a language with pleasurable accessories"
Aristotle believed in language and its ability to be embellished. In many tragic plays, choruses were used to embellish the language. Given the musical quality of diction and the chorus, the language of many plays was pleasurable.
"In a dramatic, not in a narrative form"
The play, by its nature, is dramatic. It is not a single person telling a story about a happening. Instead, it is acted out, by many people, over many times (or years).
"With incidents arousing pity and fear"
In the typical tragedy, the audience should feel a sense of pity and fear for the tragic hero and other characters. The action of the play is definitive enough that the tragic hero's character is shown as being one who needs pity (for one reason or another).
"Wherewith to accomplish a catharsis of these emotions"
A catharsis refers to a purging or purification. The catharsis allows the audience to see the tragic flaw of the tragic hero be the "end all." Given the continual rise of emotion over the course of the play, the final fall of the tragic hero allows the audience to finally come down from the heightened emotions they have felt.
Outside of his definition, Aristotle believed that art was imitation. It was through this imitation of life (through art) that people learned. Therefore, the tragedy associated with art is meant to educate the audience on the pitfalls of hamartia (or the tragic flaw).