In literature (and in theatre, in this case), "exposition" refers to the first part of the work that introduces the primary characters, the setting, and, in some cases, the main conflict. Why is act 1 of Shakespeare's The Tempest considered the exposition? Let us go over how this section of the play serves that role: how does it introduce setting? What is the conflict? Who are the main characters? What are their backstories?

Setting

The action opens on a boat that is at sea in stormy weather. Onboard is a royal party, including Alonso, the king of Naples, Sebastian, the king's brother, Ferdinand, the king's son, and Gonzalo, the king's councilor, among others.

Conflict

In the very first scene of the play during the storm at sea, there is discord between Gonzalo and the boatswain. The boatswain wants the passengers to get out of his way and stay below deck so that he and the sailors can try to save the ship (and the lives of the people on it).

Boatswain
Do you not hear him? You mar our labour: keep your cabins: you do assist the storm.
Gonzalo
Nay, good, be patient.
Boatswain
When the sea is. Hence! What cares these roarers
for the name of king? To cabin: silence! trouble us not. Gonzalo
Good, yet remember whom thou hast aboard.
Boatswain None that I more love than myself. You are a
counsellor; if you can command these elements to
silence, and work the peace of the present, we will
not hand a rope more; use your authority: if you
cannot, give thanks you have lived so long, and make
yourself ready in your cabin for the mischance of
the hour, if it so hap. Cheerly, good hearts! Out
of our way, I say.

This passage (and Gonzalo's swearing to seek revenge on the boatswain in the lines that follow) establishes a central conflict in the play. Gonzalo demands respect for the king, and the boatswain does not care. Their values are at odds, and Gonzalo is offended by the boatswain's apparent ambivalence about who exactly is onboard. The boatswain, meanwhile, holds human life—the value of any human life, his own included—above any socially constructed ideas about who is more important and who deserves to be saved or treated carefully.

Characters

We meet many characters in the first act. You asked about four in particular: Prospero, Miranda, Ariel, and Caliban.

Prospero is a character we do not encounter until the second scene, as he and his daughter, Miranda, are sitting on an island watching the boat being tossed around on the waves. Miranda asks him to make it stop—she knows that he has some special power to create the storm—and he tells her of their surprising backstory.

He tells her that he was the duke of Milan, which makes Miranda, his daughter, a princess. He trusted his brother, Antonio, to manage the rule of the kingdom, but Antonio plotted against him, ultimately ousting Prospero and sending him (and his little daughter) away from the kingdom in a little boat.

Miranda is Prospero's daughter. She falls in love at first sight with Ferdinand, which tells you plenty about her age and experience—even in the age of Shakespeare this qualifies her as youthful, idealistic, and innocent; she has not had much romantic experience so far. This fits in well with another fact we learn about her backstory in act 1. She does not know very much about her own family history. She knows that Prospero created the storm that shipwrecked the boat, but she does not know how they ended up on the island or why her father has magical powers.

Ariel is a sprite that flits around the island in the beginning of the play, reporting back to Prospero on the conditions of the boat and the people on it. After delivering the details, Ariel asks to be set free, but Prospero refuses. We understand, as readers, that Ariel is somehow obliged to Prospero:

Ariel
Is there more toil? Since thou dost give me pains,
Let me remember thee what thou hast promised,
Which is not yet perform'd me.
Prospero
How now? moody?
What is't thou canst demand?
Ariel
My liberty.
Prospero
Before the time be out? no more!

Prospero refers to a "torment" that he freed Ariel from; it has to do with the last character, Caliban.

Caliban is known in literature as a "noble savage." He is the son of a witch, and he serves as Prospero's slave. However, he is not easy to manage: Prospero has to use his own magic to control him. Prospero says that Caliban tried to rape Miranda. However, Caliban says that Prospero has been deceitful. He also says that the island belongs to him. This sets up another interesting conflict in the play—which is why this first act is considered an exposition.