Why is it said that the Duke has to "Eat his words" in Act 1, scene 3, after Brabantio accuses Othello of using witchcraft on his daughter?
When Brabantio tells the Duke of his troubles, that his own daughter has been won over by her lover through unnatural and unlawful means, the Duke is at first sympathetic to his plight.He goes so far as to tell Brabantio that he should pass sentence on the accused and have him brought to death:
Whoe'er he be that in his foul proceeding
Hath thus beguiled your daughter of herself
And you should of her, the bloody book of law
You should yourself read in the bitter letter,
After your own sense, yea, though our proper son
Stood in your action.
(lines 78-83 Folger Shakespeare Library edition)
It is when the Duke learns that it is Othello who is accused by Brabantio of such acts that his tune suddenly changes and thus, he must "eat his words." The Duke has trust in, and regard for Othello.
When Brabantio first says Desdemona has been tricked and stolen from him, the Duke says whoever tricked Desdemona will pay for it. The Duke says Brabantio himself can decide the punishment, even the death penalty--even if the criminal is the Duke's own son. However, when Brabantio says it's Othello, the Duke asks Othello what he has to say. Othello makes a speech, Brabantio again accuses, and the Duke says he needs evidence; Brabantio can't just make accusations. Basically, when the Duke finds out it's Othello (such an important military leader), he thinks twice about letting an angry father decide the punishment.