They manage payments and benefits for unemployment, disability, and Paid Family Leave (PFL), the last of which is paid time off from work you can take when you have a new baby. I think they also manage some statistics and data around employment disability etc.
PFL in California guarantees the state will pay you something like 70-80% of your typical wages, for up to 8 weeks, while you take time off to bond with and care for your baby.
There is no guaranteed paid paternity or maternity time in the USA so I am lucky to live in a state that offers this. You may view this program as either generous or paltry compared to your own countries offerings.