How does Othello’s final speech reestablish his greatness?
In his final speech, Othello requests that nothing should be "set down...in malice." From his tone, and the fact that he has acknowledged that he has "done the state some service," what he actually seems to be asking is for it to be duly recorded that it was, indeed, Othello who killed Desdemona by his own will: "nothing extenuate." In accepting his own culpability in the matter, rather than attempting to shift all blame onto Iago, whose plot has now been revealed, Othello is able to perform a final honorable act.
Othello seeks to give some explanation for what has happened—that he was "wrought perplex'd in the extreme," and that he "loved not wisely, but too well." These are a means of helping the listeners to understand his actions, but Othello does not attempt to excuse himself. When he says that he "threw a pearl away / Richer than all his tribe" and that his eyes "drop tears as fast as the Arabian trees / Their medicinal gum," he accepts, through his language, his own agency in the matter, while also expressing that he now repents of his deeds and is deeply sorry for them.
Having apologized, furthermore, Othello's final act of killing himself is one which, in Othello's eyes, seems aimed at redeeming himself. We know he would rather not go on living, as he indicates to Iago after Iago declares that he is wounded, but not killed, when Othello stabs him. But Othello begs the listeners to remember an occasion when he, once great in the field, "took by the throat" an enemy of Venice, "and smote him, thus." Just as he once dealt just retribution to this "circumcised dog," Othello now stabs himself, suggesting that he too has acted against the laws of Venice, but, also, that he intends to rectify the situation as far as he can in dealing out self-justice.
Having "done the state some service", Othello knows that the record of the events leading to Desdemona's death could be biased. Iago having been captured for the plot to murder Cassio, it is possible that Othello could be portrayed as a somewhat innocent victim of the plots of a madman. However, Othello begs against this. He wants the truth to be presented cleanly. Othello asserts that is was his own foolishness and jealousy that caused Desdemona's death, and not the actions of Iago, which should have had no effect on him. By repenting and by demanding the truth, Othello reestablishes his honorable nature, and therefore his greatness.