The literal meaning is fairly obvious. A chain can only work as long as all the links are equally strong. But if one, just one, is weak and breaks then the whole chain becomes weak. So therefore, as the saying goes - a chain is only as strong as its weakest link.
In the figurative it means pretty much the same thing - i.e. a group of people is only as strong (or as unified, cohesive, etc) as its weakest person. Once that person breaks down the whole group weakens.