The frame story is a story about someone who then tells another story, which then might contain other stories, and so on.
It is a device to unify a group of stories, but it serves another purpose. As you are reading or listening to
the story of someone telling a story, you become aware of different layers of reality -- i.e., that your reality is also "in a story".
It can give a feeling of legitimacy to a story. For example, when you hear an urban legend, it is common for it to be in a primitive frame story. "I was talking to my doctor, and he said that his cousin's wife's sister-in-law was the person who microwaved her poodle." The story seems more important because other people told hit and heard it and then told it again. If you just tell the story of how the leopard got its spots, then the story must stand on its own merits. If you hear that Margaret Thatcher was telling the story of the leopard and its spots to Queen Elizabeth, then it adds a new dimension.
Some examples of classic frame stories: