In Act III, Scene ii, how does Antony show he understands the people better than Brutus?
In Shakespeare's Julius Caesar, Antony actually shows he understands the Roman mob better than Brutus earlier than when he delivers his famous speech. In Act III:i he demonstrates his knowledge of the people when he pretends to accept the killing of Caesar as necessary for the good of Rome, when he pretends to believe that the conspirators were acting nobly when they killed Caesar, and, most importantly, when he requests permission to speak at Caesar's funeral.
That's all I seek [to be given evidence that the murder was justified];
And am moreover suitor that I may
Produce his body to the market place,
And in the pulpit, as becomes a friend,
Speak in the order [during the ceremony] of his funeral.
Antony is apparently already planning to turn the mob in his favor when he speaks at the funeral. When Brutus gives permission for Antony to speak "In the same pulpit whereto I am going,/After my speech is ended" Antony need only reply: "Be it so;/I do desire no more." Whoever speaks last will win the minds of the crowd.
During the speech you ask about, Antony fulfills the "letter of the law" that Brutus has insisted on by not directly saying anything against Brutus or the conspirators. But he manipulates the fickle Roman mob by using irony to expose the conspirators and incite the mob to violence. When he says "Brutus is an honorable man," he means the opposite. And the crowd understands.
Antony understands the Roman people are easily manipulated, and that is exactly what he does in his speech. He doesn't try to incite them against the conspirators by calling Brutus and the others names. Antony starts his speech by claiming he is "a plain blunt man" who is no match for Brutus in stirring the Roman people through speeches. By doing this, Antony gives the Romans the illusion that he is not there to cause trouble when in fact that is exactly what he's there to do. He sheds tears for Caesar, showing the crowd Caesar's will, and then telling them he cannot read it to them. Antony knows they will demand to hear it, but first he shows them Caesar's bloody robe and points out each stab mark made by each conspirator. Then Antony reads Caesar's will, and the crowd pounces on the conspirators. Antony basically puts on a show for the people, understanding how to manipulate them into a murderous frenzy. His voice drips with sarcasm as he refers to the conspirators as "honorable men", knowing the crowd will understand what he's really saying.