Because Aristotle sees society as being built out of small units of human relationships, he also sees the problems of society as being the problems of these individual relationships.
Because a large nation is composed of many parent/child units, many man/woman units, and many employer/employee units, if something starts to go wrong with the way people treat each other in those settings, it will be a problem for all of the nation. Specifically, for Aristotle, three problems that can affect society are: divorce, adultery, and illegitimacy. A marriage is a contract, a covenant, a working relationship, and a promise: if people do not fulfill their commitments to treat each other well, and to care about each other, then not only will the marriage suffer, but society as a whole will suffer, and if there are too many divorces, it will be a serious problem for the nation. Adultery is a failure to be faithful to one's spouse: too much of it will bring down a nation. If a child is born illegitimately, not only will the child face hardships, but the energy of the nation will be partially spent trying to rectify the situation, and that energy will not be available for other needs.
Remember that Aristotle discovers these principles as natural laws, like the laws of chemistry and physics. Human societies all start as monarchies, and the human tendency toward forming a state is simply part of natural human growth.