There are strong links between The Tempest and William Shakespeare's own life. These revolve around the character of Prospero, his powers, and where he is in his "career" on the island.
Start with the basics: both are men of mature age. That's basic, but what follows is less basic.
Prospero is a wizard. On this island, people do what he says. They move according to his wishes. He organizes the drama early in the play, and the narrative twists and turns throughout it. At times, his magic makes them see things that aren't really there.
Shakespeare is a playwright, not a wizard, but on stage, people do what he says. Characters move according to his wishes. He structures the drama throughout his plays, and, when he moves into his poetic moments, like in the monologues, he makes us all see things that aren't there.
At the end of the play, Prospero sets his staff (his magic) aside. This furthers the comparison between Prospero and Shakespeare because The Tempest was Shakespeare's final play.