The main way the states aid the national government is by taking care of localized tasks. State powers reduce the strain on the national government because the national government does not have to directly oversee these things. States also provide resources, such as taxes and natural resources.
The United States is set up so that no one branch or level of government has total power. The things necessary for the country to run are divided between the states and the national, or federal, government. In this way, the states help take a lot of stress off the national government. States control licenses and elections and see to the welfare, health and safety of citizens. This makes it easier for the federal government to take care of its powers, such as dealing with foreign issues.
Because states are relatively independent, they can also test things out before the national government picks it up. For example, marriage rights and the legalization of marijuana are typically under the regulation of states, even if their decisions run counter to those of the national government.
Additionally, states provide revenue, tax money, natural resources and people for the national government. These resources allow the national government to continue running.