Antony masterfully plays the crowd. He does not attack them or shout about how terrible the conspirators are. Instead, he establishes a bond with them, emphasizing that all of them are Romans, no matter how they initially feel about the assassination. However, he does put in little digs at Brutus and company, calling them "honorable men" before he lays into them.
He lists the many good things Caesar did, then follows them up with the grievances the conspirators had against Caesar. In this way, he appears to be even-handed in responding to their arguments while being rather critical indeed. Antony is mainly interested in refuting the idea that Caesar was ambitious. At one point, he stops, claiming that he is too grief-stricken to continue, giving his argument an element of pathos—he wants the crowd to believe his sincerity.
The reading of Caesar's will is what cements Antony's success. The contents of the will persuade the people that Caesar only ever had the good of Rome in mind. This, not any rage-induced tirade, is what causes them to turn against Brutus and the conspirators in the end.