In Othello, why does Iago tell Othello that Brabantio badmouthed him?
You are refering to Act I scene 2 which introduces Othello and Iago talking about Othello's marriage to Desdemona and how Brabantio is opposed to it. Let us remember what has happened before this scene. Iago has persuaded Roderigo to deliberately stir up Brabantio, alterting him to the way that Desdemona has secretly married Othello without her father's permission. If you look carefully at what Iago says to Brabantio, and the kind of animal sexual imagery he uses, he is deliberately stirring up Brabantio to cause problems for Othello. In the same way, therefore, in this scene, we see Iago trying to stir up Othello against Brabantio to make the inevitable conflict between them that more heated. Note what he tells Othello about Brabantio:
Nay, but he prated
And spoke such scurvy and provoking terms
Against your honour,
That with the little godliness I have
I did full hard forbear him.
Iago thus tries, unsuccessfully, to stir up Othello's anger against Brabantio in teh same way that he has already successfully achieved this goal with Brabantio in Act I scene 1.