What was Big Willy Shakes going for?
Antony uses a little reverse psychology on the crowd, getting them to clamor to hear Caesar's will by insisting that they shouldn't hear it. He descends to read them the private document but gets sidetracked by mourning over Caesar's body.
Again Antony insists Brutus is honorable, but then points out the gash Brutus made in his friend's bloody body. How could a friend have made such a deep wound? Antony repeats this pattern over and over (Brutus is awesome… and yet… ), until all are in agreement to burn, slay, and otherwise do not-so-nice things to Brutus and the other conspirators.
Given his wildly successful speech, we probably don't have to tell you that Antony's strong suit is rhetoric (the art of speaking persuasively). This makes him a terrific politician. In this speech, he even uses Brutus' own words to convince the Romans.
A few lines before this speech, Brutus says, "Romans, countrymen, and lovers! Hear me for my cause, and be silent, that you may hear" (3.2.13-14). Antony twists this and uses it to say Brutus and gang are not who they appear to be. Boom.
Shakespeare's genius little phrase gives Antony the power he needs to make the Romans see through Brutus' lecture. He's able to deliver a carefully crafted eulogy that's designed to turn the people against the conspirators and launch him into a position of power.
The success of Antony's speech suggests that effective leadership goes hand in hand with effective rhetoric. Antony can deliver a better speech than Brutus, so Antony is a better leader. Right after Antony finishes talking, all hell breaks loose and civil war ensues. And that's exactly what Antony intended. Mission (however suspect) accomplished.