Wouldn't you? Think if you were in the same position. Your best friend has just been murdered by a bunch of men--do you stand alone and accuse them of wrongdoing, or do you schmooze them a little and tell them they are great guys, honorable guys, and all you ask is an opportunity to speak at your friend's funeral?
Antony is an intelligent fellow. He knows his immediate safety is in jeopardy--never forget that the original proposal included Antony's murder as well. Cassius wanted them both gone, and Brutus is the one who said that Antony was like Caesar's arm when it had been cut from the body. Brutus misinterpreted Antony's power, and that is his downfall.
Antony knows that he must "kiss up" to these men in order to get them the punishment he feels they deserve for murdering Caesar. Antony knows how to manipulate, and getting these men to believe he is harmless is purely political on his part.
It's a purely political manuever. Antony is trying to figure out which way the wind is blowing, so to speak. Notice that he begs forgiveness of the dead Caesar that he is meek and gentle with these "butchers."
Antony wants to stay on their good side right now so that he can stay alive and figure out what to do next that will allow him to take control. Antony senses that the crowd, at least for the moment, is on the side of the conspirators - remember when Antony is about to give his funeral oration that the talk in the crowd is that he'd better not speak poorly of the conspirators? Antony is the consumate politician.