What happens when Miranda and Ferdinand meet each other for the first time?

In act 1, scene 2, Miranda and Ferdinand look upon each other for the first time and both characters instantly fall in love. When Miranda looks at Ferdinand, she asks her father if she is looking at "a thing divine" and says that she has never seen anyone so noble. Likewise, Ferdinand sees Miranda for the first time and asks if she is a goddess because she is so beautiful. For Miranda, Ferdinand is the first man, aside from her father and the creature Caliban, who she has seen and is taken back by his handsome appearance. Prospero recognizes that Ferdinand and Miranda have fallen in love at first sight and is pleased. However, Prospero does not want them to take their love for granted by making it too easy and instantly accuses Ferdinand of being a spy, who plans on taking the island from him. Prospero proceeds to arrest Ferdinand and makes him labor as his slave until it comes time to announce that he is the rightful Duke of Milan. Despite being Prospero's captive slave, Ferdinand takes pleasure in his labor and works towards winning Miranda's heart.

It's interesting, with this question in mind, to note just how much of the action in the tempest is itself being manipulated by Prospero, through his magic and illusion, to advance his own aims. It does seem like Miranda and Ferdinand's encounter and relationship falls into this same theme and is itself (at least on some level) under the influence of Prospero's manipulations.
The moment your question refers to comes in the later part of act 1, scene 2. Ferdinand appears, being led by an invisible Ariel, into the presence of Miranda and Prospero, and Miranda and Ferdinand fall under each other's spell. Even so, Prospero believes that, if their relationship is to become stronger, the two need to contend with real adversity. With this in mind, Prospero sets himself up as an antagonist, accuses Ferdinand of treachery, and takes him prisoner.

This is a classic Shakespeare case of love at first sight, and, in Miranda's case, love of the first not-her-dad-not-monster-Caliban male she's ever seen since she was a very small girl. As Prospero admonishes her:
"Thou think'st there is no more such shapes as he/Having seen but him and Caliban: foolish wench!/To the most of men this is a Caliban/And they to him are angels."
Miranda stands up for Ferdinand, which is an important point, since it's unlikely she's defied her father much, if at all, before this. Of course, as we quickly learn, Prospero doesn't really object to Ferdinand; he's just testing him, making sure he's worthy of Miranda's love.
As in so many instances of instant love in the Shakespeare canon, we do wonder if this magical attraction will result in a happy marriage, but with "The Tempest," we simply accept their love for each other as part of the suspension of disbelief.

When Miranda meets Ferdinand for the first time, it's love at first sight. Miranda has never seen such a handsome, dashing man before. But then again she's never clapped eyes on any man other than her father, Prospero. And as for handsome, well if you'd grown up on an island with Caliban, then just about anything would be good-looking by comparison. Ferdinand, however, has seen many beautiful young women before but none as fair as the comely Miranda. Immediately he introduces himself to Miranda as the Prince of Naples.
Prospero is pleased that his daughter and Ferdinand have fallen for each other, but he doesn't want their blossoming love to fade too quickly. So he puts another one of his crafty little tricks into effect. He accuses Ferdinand of being an impostor and threatens to have him imprisoned. Ferdinand is furious at this outrageous slur upon his honor and responds by drawing his sword. But no sword is a match for Prospero's magical powers, and he casts a spell over the hapless Ferdinand which keeps him nice and still. Prospero is then easily able to lead the bewitched Ferdinand off to his (thankfully) temporary captivity.