Apollo Data Sources` are a crucial part of building robust GraphQL APIs. They encapsulate the code for interacting with specific data services, such as databases or external APIs, and provide access to the GraphQL context. By using Apollo's built-in `DataSource` classes, developers can create custom data sources that integrate seamlessly with their application. The process involves creating a parent class that defines the policy for interacting with a particular data source, and then extending it with a child class that performs operations specific to the application. This allows developers to cache and batch data fetching, as well as implement authorization headers and error reporting. With this knowledge, developers can create high-performance GraphQL APIs that meet the needs of their users.