Next.js API routes allow developers to create server-side logic separate from the client-side, offering functionalities such as database connections without exposing sensitive information to clients. The tutorial provides a step-by-step guide on setting up API routes using file-based routing within the pages/api directory, covering static, dynamic, nested, and catch-all routes. It emphasizes the importance of securing these routes through JWT authentication, providing detailed instructions on implementing user authentication and token verification. As an alternative to manual authentication setup, the text introduces Clerk, a solution that simplifies the process by managing authentication tasks. With examples and hands-on exercises, the tutorial illustrates how Next.js API routes can enhance server-side logic integration while highlighting future changes and the potential use of Edge API routes for improved performance.