How does Iago plant the idea of Desdemona's infidelity in Othello's mind, and how does he make it grow?

Act 3, Scene 3 is the fateful moment in the play when Iago plants the first seeds of suspicion in Othello's mind. As always, Iago is remarkably subtle, using sly hints and innuendo to carry out his foul deed. At the beginning of the scene, Cassio is in conversation with Desdemona, hoping that she will intercede with her husband on his behalf to get him reinstated as Lieutenant.

Iago and Othello suddenly enter; Cassio hurriedly departs. It all looks rather suspicious, even though there is absolutely nothing going on between Cassio and Desdemona. There is a perfectly valid reason for Cassio's haste: he is still embarrassed about his drunken antics the night before. Nevertheless, Iago immediately spies his opportunity and takes it:

IAGO

Ha! I like not that.

OTHELLO

What dost thou say?

IAGO

Nothing, my lord: or if—I know not what.

OTHELLO

Was not that Cassio parted from my wife?

IAGO

Cassio, my lord! No, sure, I cannot think it,
That he would steal away so guilty-like,
Seeing you coming.

OTHELLO

I do believe 'twas he.

Notice how Iago explicitly says that he does not think that Cassio would leave "so guilty-like." Yet, at the same time, he is deliberating establishing a link between Cassio and the word "guilty," immediately sowing more than a hint of suspicion in Othello's mind. Iago's evil ploy appears to bear fruit almost at once. Othello starts acting abruptly toward Desdemona when she asks him when he might accede to Cassio's request:

Prithee, no more; let him come when he will, / I will deny thee nothing.

When they are alone together, Iago asks Othello further leading questions about Cassio, designed to fan the flames of jealousy further. Sensing Othello's growing suspicion, Iago takes things up a notch:

Good my lord, pardon me:
Though I am bound to every act of duty,
I am not bound to that all slaves are free to.
Utter my thoughts? Why, say they are vile and false;
As where's that palace whereinto foul things
Sometimes intrude not? who has a breast so pure,
But some uncleanly apprehensions
Keep leets and law-days and in session sit
With meditations lawful?

Again, Iago is using innuendo to manipulate Othello. He is implying that he has negative thoughts about Cassio, but he dare not let on what they are. He knows that Othello will compel him to reveal what he claims to know about Cassio. Othello pretends to be unconcerned, but Iago knows that is not the truth. In order to seal the deal, Iago needs some hard evidence. There is not any, of course, so he makes it up. His wife, Emilia, picks up Desdemona's dropped handkerchief, a love token from Othello, and gives it to Iago. He then plants it on Cassio to "prove" that Desdemona gave him Othello's precious gift.

By the end of act 3, Othello has completely renounced his love for Desdemona, and Iago has managed to exert complete control over him. Desdemona's days are numbered.