A story's deeper meaning can be termed its 'significance'. It represents the importance or the broader implications that the story holds. Sometimes, this significance can be related to social, cultural, or personal aspects.
The deeper meaning of a story is often called its 'moral'. It's the lesson or value that readers can take away from the story. For example, a story about kindness might have the moral that being kind leads to positive outcomes.
A story's deeper meaning is usually referred to as its 'theme'. The theme is the underlying message or idea that the story conveys.