In Act III, Scene 2 of Julius Caesar, how does Mark Antony convince the crowd to adopt his point of view?

Antony gives a meticulously crafted speech that has the sole purpose of inflaming the crowd. Antony is in a difficult position since Brutus, an excellent orator himself, has just won the crowd over by providing sensible reasons for Caesar's assassination. The crowd, therefore, is not in the mood to hear anything negative about him.

In his opening, Antony gives the crowd a reason to listen to him. He stresses their shared companionship and allegiance: "Friends, Romans, countrymen" and then uses an equivocal statement that whatever evil one commits is remembered beyond death and that one's good deeds are forgotten. In this instance, Antony could be referring to both Caesar and the conspirators. Brutus made a point of Caesar's ambition, which he deemed an evil, but it could also be argued that the conspirators' evil deed will follow them, for there was nothing good in what they did. Antony is subtly introducing a cynical element into his speech.

He then says that he is not there to sing the dead general's praises, just to bury him. When Antony refers to Brutus as "noble," he obviously intends for the crowd to support him because they believe Brutus performed an honorable deed. It will soon become apparent, however, that Antony is using verbal irony. Antony then weaves a tapestry of contrasts with his words by comparing what Brutus said to what Caesar actually did.

Once he contrasts Brutus' statement with what Caesar actually did, Antony cleverly erodes the meaning of "honorable" when referring to Brutus and the conspirators. The tone becomes sarcastic and the crowd gradually catches on.

Antony repeats that he has come to speak at his friend's funeral, indicating he had a deep love of and trust for Caesar. When Antony contrasts this with the statement that Brutus was an "honorable man," he creates doubt as to Brutus' actual intent. Antony, being so close to the deceased, would have known more about Caesar's intentions than Brutus. How then could Brutus's actions have been honorable?

In addition to creating doubt about Brutus's character, Antony really rubs it in when he tells the crowd that Caesar generously inflated the public coffers by selling slaves he had captured for ransom. Antony also states that Caesar wept for the poor, implying Caesar cares for the common folk. If Caesar was as ambitious as Brutus claimed, why would he have even cared?

The rhetorical question Antony uses is deliberately fashioned to make the crowd doubt the veracity of what Brutus said. To disqualify Brutus' assertion that Caesar was ambitious, Antony refers to the occasion during Lupercal when Antony thrice presented the crown to Caesar and he refused it: 

You all did see that on the Lupercal
I thrice presented him a kingly crown,
Which he did thrice refuse: was this ambition?
Yet Brutus says he was ambitious;
And, sure, he is an honorable man.

Antony cleverly states that he did not come to citicize what Brutus had said but that he was there to speak of what he knew, indirectly implying that Brutus did not know what he was talking about. Antony states that, with good reason, all Rome once loved Caesar. Antony asks what cause prevents the crowd from mourning Caesar, and closes his initial address by crying out that the crowd's judgment was irrational and driven by savage thoughts. Antony says this to instill guilt. He then takes a dramatic pause when he asks the crowd to excuse him, saying he is overcome by the moment—he has lost his heart and needs time to gather himself.

By this time, the crowd has great sympathy for Antony and again believes Caesar was a good leader who cared for them.

Antony then continues his address and slyly suggests that, if he were to ask the crowd to turn against Brutus and Cassius, he would be doing them a great wrong, as they are honorable men. Antony claims he would rather wrong the dead, himself, and his audience, than do Brutus and Cassius any harm. The deliberate reverse psychology is clear: Antony is, in effect, calling the crowd to mutiny.

Antony then introduces Caesar's will. 

But here's a parchment with the seal of Caesar;
I found it in his closet, 'tis his will:
Let but the commons hear this testament—
Which, pardon me, I do not mean to read—

He now knows he has the crowd in the palms of his hands. Telling the crowd that he does not want to read the will makes everyone more insistent to hear its contents. Antony says its contents are so favorable to the crowd that, if he should read Caesar's bequests, they would kiss Caesar's wounds and dip their napkins in his blood out of gratitude. The crowd cries out to Antony to read the will, but he says it would drive them to madness if they knew Caesar's legacy for them.

Antony further mentions that he is afraid he would wrong the 'honorable men' if he reveals what Caesar said in his testament. Antony deliberately mentions that the ones who stabbed Caesar would be wronged. The crowd, however, has lost its patience and insists Antony read the will.

Antony then calls the crowd closer and asks all to surround Caesar's corpse while he reads the document. He then dramatically points out the dagger wounds on Caesar's corpse, reminding the crowd of how foul the assassination was. The citizens express their grief and then passionately cry out for revenge, but Antony stays them.

The crowd has become like a ravenous beast of prey, ready to strike. Antony reads the conditions of Caesar's will. At the end of it all, the multitude carries off Caesar's corpse and goes on a murderous rampage, seeking out the conspirators. Antony, who is satisfied with the devastating effect his speech had on the crowd, remarks:

Now let it work. Mischief, thou art afoot,
Take thou what course thou wilt!