In Act I scene 2, which is when Prospero chooses to reveal his background to his daughter, and helpfully to the audience, we learn about the facts of his exile and how he had to leave his kingdom and take up residence on this strange island where the play is set. However, what is clear is that although Prospero has been living on this island for a number of years, and has claimed lordship over it, he still thinks greatly of how he was betrayed by his brother and seeks some way of gaining revenge, or at least some form of redress. Note what he says to Miranda:
Know thus far forth
By accident most strange, bountiful Fortune,
Now my dear lady, hath mine enemies
Brought to this shore; and by my prescience
I find my zenith doth depend upon
A most auspiciouis star, whose influence
If now I court not, but omit, my fortunes
Will ever after droop.
Prospero recognises that the arrival of the ship with its passengers, including his brother, gives him a chance to change his fortunes. The way he refers to these men as his "enemies" shows he remembers incredibly keenly the experience of being deposed and exiled, and is eager to do something about it. This is why Prospero, using his magical talents, effectively stagemanages the island, bringing his brother to a position where he admits that he regrets exiling his brother, before finally revealing himself and being able to return to his former position with his daughter.