I want to understand what Chandler means when he says he's maxed out after thinking he's embarrassed by his bunny costume.
He's reached his limit or maximum. He's exhausted and therefore impervious to embarrassment (not the other way around, i.e., he's embarrassed and therefore maxed out).
to reach the limit of one's capacity, endurance, etc.
I think the above definition may fit, though more details about the dialogue may help to understand the context.