Antony asks that they allow him to take the body to the marketplace and, further, that he be allowed to orate at the funeral.
Here are the lines Antony delivers to the Servant (who is to take the news to the conspirators):
Post back with speed and tell him what hath chanced.
Here is a mourning Rome, a dangerous Rome,
No Rome of safety for Octavius yet;
Hie hence, and tell him so. Yet stay awhile,
Thou shalt not back till I have borne this corpse
Into the market-place. There shall I try,
In my oration, how the people take
issue of these bloody men,
According to the which thou shalt discourse
To young Octavius of the state of things.
Lend me your hand.
When Mark Antony enters the scene after Caesar's death, the very first thing he asks the conspirators is if they plan to kill him, too. He also shows his devotion to Caesar by asserting that there would be no better place for him to die than by the side of Caesar and no better means for him to die than by "those your swords, made rich / With the most noble blood of all this world" (3.1.155-56).
After Brutus replies by arguing that they committed the bloody deed for the good of all of Rome and have no desire to likewise harm Anthony, Mark Anthony next shakes the hand of each conspirator, as a gesture of making peace, and asks them to further explain the reasons why they killed Caesar. Yet, we soon realize he shakes their hands with the purpose of manipulating them and the situation. After shaking their hands, he asks permission to present Caesar's body to the public in the market place; he then asks permission to speak on Caesar's behalf as a friend at a funeral. Since Brutus has been put at ease by Antony's gesture of peace, he willingly permits Antony to do so despite Cassius's warning that Antony could turn the public against them.
In Antony's soliloquy towards the end of the scene, we learn that Antony has every intention of turning the public against the conspirators. He vows to Caesar that Caesar will be revenged in the form of "civil strife [that] / Shall cumber all the parts of Italy" (264-65). We can interpret the phrase "civil strife" as referring to civil war. He continues further to prophecy that this civil war will be the bloodiest all of Italy has ever seen.
Antony entered a while after the conspirators had completed their deed. Although they didn’t trust him, Brutus had made an agreement with the others not to kill him. On meeting the conspirators Antony was sure he was going to be killed because of his friendship with Caesar. He however is not afraid and asks the conspirators to kill him if they deem it fit. Brutus instead tries to explain their actions and even offers Antony friendship. After Antony figured that they were not going to kill him he asked that he is allowed to shake the hands of each of them. Antony asked if he would be allowed to take Caesar’s body to the market and speak as a friend about his funeral. However the conspirators did not allow him to talk first because of their fear; instead Brutus spoke first to at least get an opportunity to shape the message to the public. In this regard, Cassius was still doubtful about allowing Antony to speak.