In any democracy, the people have the ability to participate in their government. In a federal system, they simply have more opportunities to do so. This is because the government of their state (or province or whatever the smaller units of government are called) is more accessible to them.
The national government of a country is typically a large entity that is based in a faraway city. Ordinary people have little chance to participate in it for that reason. The state government, by contrast, tends to be much smaller and much closer to home. The people who represent you in a state government represent a much smaller number of people than those who represent you in the national government. For this reason, you are much more likely to be able to have meaningful interactions with your state legislator than with someone representing you in the national legislature.
Federalism, then, gives people more opportunities to interact with a government that is closer to them. This means they are more likely to have a chance to participate in their government in a federal system.