Office politics
Iago is passed over for a promotion in the first eight lines of the play. Everything after that is a case study in how much damage one motivated person can do without ever saying anything they could be held to.
The grievance is specific, and by his own account it is not about competence.
He had sponsors. Three senior people advocated for him directly. He was passed over anyway, and the reason he was given was, in his telling, a lot of impressive language that did not amount to a reason. I know my price. Whatever else Iago is, he is a man who was told no without being told why, by a boss who preferred sounding decisive to being clear.
None of this excuses him. It does explain why the play opens where it opens, and it is worth noticing that Othello never learns any of this is a problem until it is far too late.
The method
What makes Iago dangerous is not the anger. It is that he decides, immediately, to keep working there.
I follow him to serve my turn upon him. He will be visibly compliant and privately opposed, and he says so in the first scene so the audience is never in doubt. The knee-crooking knave passage is his justification: the loyal ones get used up and discarded, so loyalty is for fools. It is a coherent worldview. It is also, note, a description of exactly the treatment he believes he just received.
He never makes a claim
The technical achievement of the play is that Iago destroys Othello almost entirely through questions, hesitations, and advice against the very suspicion he is planting.
Read that as a transcript and it is a warning against jealousy. It is the opposite. He is naming the emotion, making it vivid, and attaching it to Othello's situation while appearing to counsel against it. If anyone reviewed this conversation afterward, Iago advised his commander to avoid jealousy. That is the whole trick, and it is repeated for two acts.
He almost never asserts. He implies, then withdraws, then lets the other person supply the accusation and defends against having made it. By the time Othello acts, every damaging conclusion in his head is one he arrived at himself, which is precisely why he cannot be argued out of it.
What the play is actually about
It is easy to read Othello as a play about a villain, and Iago is one. But the mechanism only works because of the conditions around it. A promotion decision was made and never explained. A talented, resentful subordinate was kept close with no channel for the resentment. And the man in charge trusted reputation over evidence, right up until he trusted the wrong reputation.
Iago supplies the malice. The organization supplies everything else.