Coal - the black rocks that you put in the fireplace. It was seen as a job for servants and slaves not for people of status. If you were seen to do such a job, you would be humiliated. Today, "take their garbage" is the closest in sentiment (not meaning). You won't take any nonsense from them.

to carry coals = to be put upon; to submit to degradation or humiliation; to put up with insults;