Promises in Swift allows developers to write cleaner asynchronous code by avoiding callback hell. By using PromiseKit, developers can easily create promises that resolve or reject with values, making it easier to handle errors and chain functions together. The article demonstrates how to use Promises to fetch a random GIF from the Giphy API, download the image, and display it in a UIImageView, all while handling errors and exceptions cleanly. With this approach, developers can write more readable and maintainable code, making it easier to extend into more complex scenarios.