What does Antony convince the crowd of in his funeral speech?

Before Antony addresses the mob, he speaks an eloquent soliloquy ostensibly addressed to the dead Caesar. He reveals his intention of betraying Brutus and the others and causing a widespread riot which will drive all the conspirators out of Rome.

Over thy wounds now do I prophesy
.........................................................
A curse shall light upon the limbs of men;
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And Caesar's spirit ranging for revenge,
With Ate by his side come hot from hell,
Shall in these confines with a monarch's voice
Cry “Havoc!” and let slip the dogs of war,   III.1
                                                    

Antony is all alone, and his life is in danger. He feels that his best prospect is to start a general riot, using Caesar's wounds and Caesar's will as his best weapons. If the conspirators are forced to flee the city, that will not mean they are defeated. Brutus and Cassius will raise armies, and there will be chaos throughout Italy for years. Antony intends to try to start a mutiny by persuading the mob attending the funeral that they were all personally injured by the conspirators because Caesar was their best friend and the conspirators committed treason when they butchered him. Antony's thesis might be summed up in the following excerpt from his funeral speech.

Even at the base of Pompey's statue,
Which all the while ran blood, great Caesar fell.
O, what a fall was there, my countrymen!
Then I, and you, and all of us fell down,
Whilst bloody treason flourish'd over us.    III.2

Shakespeare has Antony speak the soliloquy beginning with 

O pardon me, thou bleeding piece of earth,
That I am meek and gentle with these butchers...

in order to show the audience that Antony is not just an athlete and an ignorant soldier, but also a highly intelligent and eloquent man. Otherwise his funeral speech would come as a big surprise. The audience knows that Antony is eloquent, but Brutus does not, or he would not have let Antony speak. Antony's funeral speech is probably the best thing Shakespeare ever wrote. It completely turns the tables on Brutus and Cassius. They are forced to ride out of Rome "like madmen." Antony is successful in triggering a city-wide riot involving arson, looting, and murders. He shows that he has accomplished what he intended when he says:

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

We have seen instances of urban mob rioting in America in recent times. Mobs always behave the same way. The first thing they think of is setting fires. This is what first occurs to the Roman mob.

We'll burn his body in the holy place
And with the brands fire the traitors' houses.

Since they are leaderless, they are capable of all kinds of senseless acts of destruction and violence. Shakespeare illustrates this mob-madness in Act III, Scene 3, in which a group of rioters encounter Cinna the Poet and tear him to pieces for no reason. The people at the bottom of society are chronically discontented. Men like Antony have always used the lowest class for their own purposes. Antony doesn't tell the mob what to do. He doesn't care what they do as long as they create total chaos. Eventually Antony, Octavius and Lepidus will take control and quell the worst disorder. But they themselves are in jeopardy. Their triumvirate is unpopular and unstable. In Act IV, Scene 2, Antony tells Octavius:

Brutus and Cassius
Are levying powers; we must straight make head;
Therefore let our alliance be combined,
Our best friends made, our means stretch'd;
And let us presently go sit in council,
How covert matters may be best disclosed,
And open perils surest answered.

To which Octavius replies:

Let us do so, for we are at the stake,
And bay'd about with many enemies;
And some that smile have in their hearts, I fear,
Millions of mischiefs.

Antony fully intended to start a riot by convincing the crowd that they had been betrayed and victimized by Brutus, Cassius and all the other envious conspirators. Antony had no alternative. He could not foresee how such a mob riot would end, only how it would begin. The men who were listening to his speech carried their outrage to others who had not heard Antony's speech. The rioting spread from quarter to quarter like a fire. Order would not be restored until some time after Brutus and Cassius were defeated at the Battle of Philippi, which is dramatized in Act V.