Prospero listens to Miranda and Ferdinand so he can tell if they are falling in love.
Prospero was betrayed by his brother, Antonio, and Ferdinand’s father, Alonso. They took his kingdom from him and stuck him on a boat. He ended up on this island. As fate would have it, Antonio and Alonso strayed into his path, and he used his magical powers and accomplices to shipwreck them on his island.
Ferdinand was separated from the rest of the ship, and Alonso thinks he is dead. Prospero sees Ferdinand as the perfect opportunity to exact some revenge on Alonso by doing some matchmaking. He hopes to make Ferdinand and Miranda fall in love. He spies on them to make sure that it works.
Prospero wants to make sure that Ferdinand has to work for Miranda.
They are both in either's powers; but this swift business
I must uneasy make, lest too light winning
Make the prize light. (Act 1, Scene 2)
In other words, if he doesn't work for her he may not value her. Thus, Ferdinand has to do some chores in order to win Prospero’s trust. Miranda has never seen any young men, and she is infatuated with him. She sneaks over to him and talks to him. Prospero is aware of everything, but they have no idea.
My mistress, dearest;
And I thus humble ever.
My husband, then?
Ay, with a heart as willing
As bondage e'er of freedom: here's my hand.
And mine, with my heart in't; and now farewell
Till half an hour hence. (Act 3, Scene 1)
After they leave, Prospero remarks that he is just as happy as the young couple at this development, but he is too busy to worry about it right now. They do not know that he is calculating and surveilling their every move, and that it is by his design that they are matched.