What does Antony want from the conspirators in Julius Caesar?

Antony wants the conspirators to let him speak at Caesar’s funeral.

Mark Antony was a trusted member of Caesar’s army.  He was apparently dangerous enough that the conspirators made sure that he wasn’t with Caesar at the capital during the assassination after Brutus insisted he be left alive. 

When Antony sees conspirators above Caesar’s body, he does not attack them.  In fact, he is immediately conciliatory.

I do beseech ye, if you bear me hard,
Now, whilst your purpled hands do reek and smoke,
Fulfil your pleasure. Live a thousand years,
I shall not find myself so apt to die … (Act 3, Scene 1)

Antony tells them that if he is going to die, he would rather die next to Caesar.  Brutus tells Antony that they are not going to kill him.  Antony gets a little rowdier at this point, telling them Caesar was a great man and asking them to give reasons why he was dangerous.  He then asks for something else.

ANTONY

That's all I seek:
And am moreover suitor that I may
Produce his body to the market-place;
And in the pulpit, as becomes a friend,
Speak in the order of his funeral. (Act 3, Scene 1)

Brutus agrees to both of these requests.  Cassius, who already admonished Antony once during his diatribe, now tries to talk Brutus out of letting Antony speak.  He knows that Brutus is idealistic and naïve.  He is aware that Antony is cunning and has something up his sleeve.  He worries that if he speaks to the people at Caesar's funeral, they don’t know “how much the people may be moved.”

Brutus doesn’t listen.  He assumes that things will be fine because Antony will talk after he does.  He earlier stated that Antony wasn’t dangerous without Caesar, and he overrules Cassius now.  Antony has been subservient to him while he has the upper hand, and that is enough for him.  It is the biggest mistake he could make, and the turning point for him.  Brutus was on top when he killed Caesar, but he won’t be for long.