끈 is a cord, tether, or string (구두끈 = shoestring, 가방끈 = strap of a bag). By extension it can also mean something by which you hold on to something important, like a connection, lifeline and so on, similar to such English expressions as 'hanging by A THREAD', 'at the end of MY ROPE', etc. So 끈(이) 떨어졌다 is like the rope that you were hanging by for survival is severed, meaning you're in a desperate situation. It is often about making a living or even finding the next meal just to survive (the dictionary says in some dialects 끈 also means 'finding a meal'). 둘 다 끈 떨어진 처지이지만 is thus "although we're both in a situation/bind with no more rope left (i.e. facing a dire dituation)". 둘 다 끈 떨어진 처지지만 조금 더 상황이 나은 그가 보내는 마지막 도움인 것이다. = It is the last offer of help from him, whose situation is somewhat better than mine, even though we're both at the very end of our ropes.