In Act 3 Scene 3, What 'proof' does Iago eventually provide for Othello?

He doesn't provide proof, he provides suspicion, but for some suspicion becomes truth.

Iago recounts: Cassio supposedly called out to Desdemona in his sleep, telling her to be cautious and hide their love. Then, Cassio started writhing around in the bed and kissing Iago's hand as if it were Desdemona. At this point, we're kind of confused as to why Iago wouldn't wake Cassio up and say, "Excuse me, please stop making out with my hand." Anyway, the still-asleep-and-dreaming Cassio then supposedly threw his legs over Iago's thighs, kissed some more, and finally declared, "Cursed fate that gave thee to the Moor!" (3.3.426), in case Iago had left any doubt.

Iago, with great timing, puts that final nail in the coffin. Still playing innocent, and instructing Othello to be calm (which only helps his fury), he tells Othello he saw Cassio with Desdemona's special handkerchief.

Hearing this, Othello announces all his love for Desdemona is gone. He's now out for blood and revenge, hopefully in one convenient package.