It's also a nice revenue generator for the state. Many states collect child support payments if the mother is on public assistance. They don't particulary care what the facts are; what's important is to find some man to extort money from. What's more, they're empowered by the Congress to reach across state lines to get their dough.
It is a revenue generator, under the guise of welfare reform. And the federal government pays 2 dollars for every 3 they collect. So if the man pays 200 through the state governement, 133 dollars is paid from the federal coffers to the state. When you add in the costs of collecting support, and the bad collection rates (often these rates are held up by the guy who would pay directly to the mother on time every month but pays through the state for accounting) I doubt it saves any money at all.