How do you explain the relationship between a country and a state?

The words country and nation are often interchangeable. A nation is a sovereign or independent state. Think of Louis XIV's famous statement: "I am the state". He was proclaiming that his royal person symbolically defined the sovereign state of France. The US started out essentially as 13 independent nations, or states. These states entered into an agreement, The Constitution, that brought the states together under a federal government. There is often some tension or interplay between the US states and the federal government when there is a question whether a given power or authority belongs to the states or whether it has been given to the federal government. This idea of sovereign state is why that tension exists. The former Union of Soviet Socialist Republics was structured in a loosely similar way. There were several Soviet Socialist Republics that were brought together under the Kremlin.

So in the US there may be, understandably, a general belief that a state is a section of a country or nation. But on the world stage, state, sovereign state and nation are much closer to the same in meaning.