This is a great question. Antony does three things to move the audience at Caesar's funeral. First, he appeals to Caesar's generosity. At first he does admit that Caesar was ambitious, but he also says that he was generous. He loved the poor and even wept for his own countrymen. For example, Antony states: “When that the poor have cried, Caesar hath wept” (III.ii.88)
Antony also plays on Caesar's love. Caesar truly loved his people and benefited Rome with great wealth. At one point he even says that he would read Caesar's will. When he finally does, it is made known that left everyone in Rome money and decided to build parks and other thing to benefit Rome. All of this moves the plebeians and makes them weep.
Finally, Antony downplays eveything from a rhetorical point. He says that he is not an orator like Brutus. Moreover, he calls Brutus and Cassius honorable men so many times. All of this is ironic, because he is playing with the the audiences emotions. Why? What Brutus and Cassius has done is not honorable, at least according to Antony. Antony wins over the crowd.