GraphQL enables schema stitching by creating a single schema from multiple underlying APIs, allowing developers to merge different databases such as Contentful and Postgres into a unified GraphQL API. This is achieved through remote schema relationships in Hasura, which establishes connections between the two databases. By setting up a remote schema relationship, developers can create a single query that retrieves user data from Hasura and content from Contentful, eliminating the need for separate queries to each database. The process involves creating a new table on Hasura, establishing a connection with Contentful through an API key and authorization header, mapping fields between the two databases, and defining a mutation to set favorite recipes. With this setup, developers can build a unified backend API using GraphQL, which can be further expanded upon for a full application.