One role of the state (by which I mean the government) is to try to ensure that society will remain stable and prosperous. It is in this way that the family is important to the state. The family is one of the basic building blocks of social stability and prosperity.
Families are the major source of socialization for children. It is the family that teaches children what society expects of them. If the family did not raise children to be basically law abiding and productive, there would be more social problems and less stability than there are now.
Being in families also tends to help adults. Adults in families are more content with their lives. Because of this, they are much less likely to go out and commit crimes or otherwise make trouble. They are also more likely to be mentally stable enough to hold jobs and do well at those jobs.
In these ways, the strength of the family supports the society as a whole. This means that families are important to the state.