The state government plays an important role in the administration and management of the country. There are state and federal offices, courts and political positions, and while some very important and significant issues are managed at a federal level, a lot of other roles are managed at the state level.
State officials are elected in local elections, and are responsible for the managing of the local area. They may implement federal law, and in some cases each state may have its own laws and its own policies which differ from or build upon the rulings that are passed down federally. This is why there are different rules about things such as firearms or alcohol consumption in different states.
State governments charge taxes and use their own budgets to cover a lot of state services (fire and rescue, street cleaning, etc). Certain benefits and housing projects are run on a state by state basis as well, as are a lot of schools. There are some private companies which offer services – again, the breakdown for this varies depending on the state and how they operate.
Some states offer things like discounted phones for the unemployed with free plans that allow them to get online, making it easier to access education and employment opportunities. Because state government offices have so much free reign provision can vary massively.
State government elections are not the same as presidential elections. It is important that you understand this and are aware of what you are voting for and who you are voting for when you take part in a local election. In many cases, the person you want running your state may not be the same as the party that you want to actually run the country; local policies matter a lot.