The design of Propel's GraphQL schema for an analytics API platform involves five key principles: starting with what developers need to build and working backward, writing the docs first, having a low barrier to entry but high ceiling, using specific over general names, and meaningful mutations. These principles aim to simplify the development process while providing flexibility and scalability. The schema design also considers evolving without breaking changes, unions on inputs, modeling analytic metrics in the schema, creating top-level metrics, and leveraging Propel as an analytics backend. By following these principles and examples, developers can create a scalable and maintainable GraphQL schema for their own analytics applications.