Why does Antony say "The evil that men do lives after them;" ? Or in other words exactly what does this statement mean?
Mark Antony is alluding to the fact that evil deeds in history are often more easily remembered than the good ones. A cursory dip into any history book will confirm this. One of the purposes of Mark Antony's speech is to mitigate any evils that Caesar may have committed while highlighting the good that he did.
Mark Antony's immediate audience consists largely of plebs, who, for the most part, loved Caesar. Indeed, it was their love for him which stirred up the ire of Brutus and the other conspirators. They hated Caesar for pandering to the mob, which they saw as an example of his overweening ambition, expressive of an ultimate desire to crown himself a king. As his audience is sympathetic, Mark Antony uses pathos, or an appeal to the emotions, to rekindle that love of the people for their fallen leader, as it may soon fade in the enveloping confusion of a rapidly-changing political situation.
It is often said that history is written by the victors. In the immediate aftermath of Caesar's assassination it is the conspirators who are the victors. And as such they are keen to rewrite the history of Caesar's rule to their advantage. Mark Antony's speech is a subtle attempt to stop them from doing this. Not only is he absolutely determined to ensure that Caesar's good name will live on, he is also going to do whatever he can to make sure that the evil of the conspirators, their bloody act of treachery, will not only be punished, but never forgotten.
Marc Antony employs many statements with hidden meanings in his famous funeral oration of Act III, Scene 2, and "The evil that men do lives after them" is certainly one of them. This statement implies that history records the wrongs of people in more inflammatory words that are long remembered, while often their good deeds are either mitigated in the shadow of the more interesting evils or even forgotten.
Those who hated Caesar were eager to speak of his evil, Marc Antony hints in his statement; this act of suggestion is his subtle way of beginning to cast aspersions upon the conspirators. Soon afterward, Antony alludes to the accusations of Brutus and the others, but he again is subtle as he adds,
The noble Brutus
Hath told you Caesar was ambitious.
If it were so, it was a grievous fault,
And grievously hath Caesar answer [for]it. (3.2.79-80)
Thus, Brutus begins to sow the seeds of doubt into the minds of the plebeians who listen, so that when he reaches the end of his speech, the crowd will be eager for rebellion.
Antony speaks these words in Act III, Scene 2 when he is giving a speech at Caesar's funeral. He is speaking ironically as he delivers this speech.
What he is saying here is that people only remember the bad things that other people have done. He says that we only remember the evil that the dead did and we forget the good. I do not know that this is true, but it is what he says.
He goes on to say that this is what they should do with Caesar. That's where the irony comes in -- he is really trying to get the people to remember the good about Caesar.