To deliver Caesar's eulogy (funeral speech)
That he be allowed to take Caersar's body to the marketplace and make a speeche to the commoners
well, because for one Antony is Caesar's friend and for two he wants revenge on the conspirators for killing his best friend. So to get revenge on the conspirators Antony is going to get the crowd to turn against the conspirators and get back at them for murdering Caesar.
To deliver Caesar's eulogy (funeral speech)
The favor that Antony asks of the conspirators is to speak at Caesar's funeral. They eventually let him, however Brutus gives him guidelines/restrictions that he must agree too or else he will not be allowed to speak at the funeral.
Antony
I'n
The reason mark Antony shakes the conspirators hands is because it's showing the death bond they have just made.
I'n
get money gang
they did not kill mark Antony and he acted to get reveng by killing the conspirators.
I'n
The conspirators did not attack Antony because of fear. This is from a book called The Tragedy of Julius Caesar Act 2.