Antony is a master at rhetoric, meaning he is a master at the art of persuasion. He was very close to Caesar and is very angry and grieved that he was killed. He is very angry, too, that Brutus, of all people, participated in the assassination plot. He knows that Brutus was Caesar's special friend, as evidenced by Caesar's surprise when Brutus stabs him ("et tu, Brutus"), and how he asks "and you [of all people], Brutus [how could you?]," which Caesar says as he is dying.
Antony is furious that Brutus has done such a dastardly thing as participate in murdering his close friend and even angrier that he now has swayed the crowd into believing he did it for honorable reasons. Antony is furious that the crowd sees Brutus as a hero who has saved Rome from a tyrant. He particularly singles out Brutus, because he is angriest at Brutus for being Caesar's friend and yet participating in the assassination. Therefore, Antony wants the crowd to turn on Brutus. He tells them, in words that have become famous in English literature, that Brutus's stab wound was the "unkindest cut of all."
Because Antony is so good at persuasion, he succeeds: the crowd does turn on Brutus. A civil war is the outcome of this assassination. Brutus should never have allowed Antony to speak, but, as always, Brutus tries to have it both ways. As the play shows, it doesn't work to try to be a nice guy sometimes and a murderous traitor other times.