To be in a delicate situation; to be on the edge of danger or ruin.
An 'eggshell' is the thin, white outer coating of an egg. The shell is very thin and breaks easily. So if you are 'walking on eggshells' you are in a situation where you could break something (or get in trouble or ruin everything) very easily.
To be out of options or alternative courses of action; to be stuck in a bad situation.
A 'rope' is thrown to someone who is in a difficult place, such as deep water or the edge of a cliff. If there is not enough rope, the person might be in trouble. So to be 'at the end of your rope' means that there is no more help available, and the situation is not good.