The tutorial explains how to use an API with Vue.js by creating a basic skeleton of a web application, defining a basic Vue.js component, creating a stylesheet for the component, adding an API call to the component, displaying API data in the component, and styling the component. The example uses the Jokes Database API to fetch and display jokes on a web page. The tutorial covers key points such as Vue.js components, lifecycle hooks, and rendering directives, providing a comprehensive guide for building APIs with Vue.js.