A more thorough explanation:

Adequate and independent state grounds is a term used by the Supreme Court to decide whether or not to hear a case from a state court. The Supreme Court will only hear a case if the state court decision was based on federal grounds and was overturned. If the state court decision was based on nonfederal grounds that are adequate and independent, the Supreme Court will not hear the case.

In the case of Republican Nat. Committee v. Burton, the Supreme Court refused to hear the case because the state court decision was based on adequate and independent nonfederal grounds. This means that the state court decision was not based on federal law and was sufficient on its own.

For example, if a state court decision was based on a state law that was not in conflict with federal law, the Supreme Court would not hear the case because the state law was adequate and independent.