What is meant when someone says the united states is a nation governed by the "rule of law"?

This means tha the United States are a political system where legal codes are publicaly disclosed and transparent and everyone is equal under the rule of law. It basically shows that there is an instance higher than public judgement or the will of an individual. All democracies are governed by the rule of law.

Those who write and enforce laws must obey the law, too.