What does "consume an API" mean?

Consuming an API here means creating a client which can send requests to the API that you build.

It appears that you need to create and API which can handle Create, retrieve, update and delete (CRUD) of a resource. For instance if your REST api is to create a blog, your API should handle CRUD functions for the object/resource blogpost. POST - Create a blog post GET - Retrieve a blog post PUT - Update a blog post DELETE - Delete a blog post.

To consume an API means to basically use any part of it from your application.