The author of this text has created a REST API using Ratpack, a Java framework, to manage users through the Couchbase repository. The API supports HTTP verbs GET, PUT, and DELETE, as well as content-type headers. The author uses Ratpack's Chain object to compose handlers that are executed in order, and Guice for dependency injection. A custom RendererSupport class is used to render User objects based on the content-type of the request. The API is built using RxJava and Couchbase's AsyncRepository to interact with the database. The text provides a detailed explanation of how to create such an API with Ratpack, Couchbase, and RxJava, including examples of how to handle different HTTP verbs and render User objects based on the content-type of the request.