Ancient Greece was a true and exagerated democracy. Everyone had to vote on everything (unless you were a woman or a slave). But many bussy citizens became fed up with having to go and vote every time on civic matters, and stopped going to the less important votes.
On days when important issues were going to be voted on, slaves would be sent out with lengths of rope dipped in paint or tar, and would slash any man who was on the street rather than taking part in the vote. This way, their togas were marked and everybody could see they had dodge their responsabilities. Hence the expression to be "roped" into doing something.