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. 

FERDINAND

My mistress, dearest;
And I thus humble ever.

MIRANDA

My husband, then?

FERDINAND

Ay, with a heart as willing
As bondage e'er of freedom: here's my hand.

MIRANDA

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.

Why did Prospero stage the pageant for Ferdinand and Miranda in The Tempest?

In act 4, scene 1 of Shakespeare's The Tempest, Prospero commands Ariel to summon spirits to perform a pageant, or masque—a courtly theatrical entertainment—for Ferdinand and Miranda in honor of their engagement.

Masques were usually performed by masked courtiers representing allegorical or mythological figures. In Prospero's pageant, the ancient Roman goddesses Juno (queen of the gods), Iris (goddess of the rainbow and Juno's messenger), and Ceres (goddess of fertility) are portrayed by spirits and Ariel rather than courtiers.

Ceres, played by Ariel, and Iris appear first (with Juno in the background), and they wish the happy couple true love and prosperity.

IRIS: A contract of true love to celebrate;
And some donation freely to estate
On the blest lovers. (4.1.93–95)

Music is heard, and Juno makes a grand entrance. Juno and Ceres sing to Ferdinand and Miranda. Juno wishes them honor, riches, and long lives, and Ceres wishes them the earth's bounty, prosperity, and a life of plenty.

JUNO: Honour, riches, marriage-blessing,
Long continuance, and increasing,
Hourly joys be still upon you!
Juno sings her blessings upon you.

CERES: Earth's increase, foison plenty,
Barns and garners never empty,
Vines and clust'ring bunches growing,
Plants and goodly burden bowing;
Spring come to you at the farthest
In the very end of harvest!
Scarcity and want shall shun you,
Ceres’ blessing so is on you. (4.1.118–129)

Juno and Ceres send Iris for the dancers—a common element of masques—but just as the dance begins, Prospero remembers that the hour is approaching that Caliban will make an attempt on Prospero's life, and he suddenly calls an end to the pageant.

In order to calm Ferdinand and Miranda for his burst of anger, Prospero makes a speech which contains two very famous lines: "Our revels are now ended," and "We are such stuff as dream are made on":

PROSPERO: Our revels now are ended. These our actors,
As I foretold you, were all spirits, and
Are melted into air, into thin air;
And, like the baseless fabric of this vision,
The cloud-capped towers, the gorgeous palaces,
The solemn temples, the great globe itself,
Yea, all which it inherit, shall dissolve,
And, like this insubstantial pageant faded,
Leave not a rack behind. We are such stuff
As dreams are made on, and our little life
Is rounded with a sleep . . . (4.1.166–176)

Prospero's intention in presenting the masque to Ferdinand and Miranda was to remind them, through the goddesses, that marriage is a blessing, part of the natural order of things, and an integral part of nature itself.

Notably, the goddesses Juno and Ceres make little mention of love or sexuality in marriage and instead instruct Ferdinand and Miranda to avoid lustful situations, to honor and sanctify their marriage and to take their proper place in the social order.