What were the circumstances of the exile of Prospero in The Tempest?

Prospero is unceremoniously ditched as Duke of Milan by a conspiracy involving his brother Antonio. The conspirators believe that Prospero is spending too much time studying the liberal arts—not to mention magic—instead of doing what he's supposed to be doing, which is governing his dukedom.
Prior to his exile, Prospero allows Antonio to manage the dukedom's day-to-day affairs. But Antonio's none too pleased at what he sees as a raw deal. He thinks it incredibly unfair that he should have all the responsibilities that come with being Duke of Milan but without the actual power and authority that the title entails. So he readily agrees to go along with a plot to usurp Prospero and send him into exile.
Prospero will still have his own little dukedom, complete with his own subjects—Ariel and Caliban—but it will be far away from Milan, on a remote little island in the middle of nowhere. At least here Prospero will be able to spend more time with his books and devote more time to honing his magic skills, which he intends to use against the men who removed him from power.

Prospero is the central character in the play The Tempest by William Shakespeare. He is a powerful magician and a scholar who lives on an island with his daughter, Miranda. Twelve years before the events of the play, he was the Duke of Milan. Then, in an act of betrayal, his younger brother, Antonio, usurped him and took his title and property.
Prospero was exiled because he gave away too much of his power to his younger brother, because he was too interested in books and in magic than in the land he was supposed to be looking after. It was when Prospero had his nose buried in his books, in his extensive library, that Antonio plotted against him, stole his title, and got him kicked out of Italy.
Prospero allowed Antonio to manage affairs of state so that he wouldn't have to and so he'd be able to continue his studies. Antonio was soon influenced by the money and power given to him and began believing he should be the duke. Prospero was so interested in learning new things, particularly learning magic, that he didn't notice his brother plotting against him.

I, thus neglecting worldly ends, all dedicatedTo closeness and the bettering of my mindWith that which, but by being so retired,O'er-prized all popular rate, in my false brotherAwaked an evil nature. (1.2.89-93)

Alonso, the king of Naples, helped Antonio in deposing Prospero. Antonio and Alonso set Prospero adrift on a boat with Prospero's young daughter, Miranda, and the boat drifted to the island where the majority of the play takes place.