What events cause Prospero to reveal his past to Miranda?

Prospero tells Miranda about his past because Miranda witnesses the shipwreck, from act 1, scene 1, in the storm and realizes it was Prospero's magic that caused it. She begs him to stop.

If by your art, my dearest father, you have
Put the wild waters in this roar, allay them.

Miranda wants Prospero to make sure that no one on the ship was harmed, but Prospero assures her that no one was. She begins to prod him about his past since he never told his full story to her.
Prospero's revealing of his past to Miranda serves multiple purposes in Shakespeare's play. First of all, it the serves the audience by providing Prospero's backstory. Prospero does this twice, once with Miranda and once with Ariel. When Miranda says, "More to know. Did never meddle with my thoughts," she is leading him to tell her more by saying that she never knew there was more to know about him. Prospero's response with "'Tis time, I should inform thee farther" is the start of what is basically a long monologue to the audience.
The second purpose of Prospero's retelling of his past is to show that power and manipulation are a constant thread throughout his history; he works the political aspects of relationships through manipulation and magic. This must be established early on in the play in order to establish the stakes of the play and to inform Prospero's magical manipulation of the storm against his enemies. By telling his own story in so much detail to Miranda, and again later to Ariel, Prospero is able to control his own story, and, thus, he is able to control the way that people perceive him and his actions.