In the blog post, Gleb Bahmutov provides a detailed guide on how to test the Browser Notification API using Cypress, focusing on a small web application that displays notification popups if the user permits them. The article explains the process of setting up tests in Cypress to verify various scenarios, such as checking if the browser supports notifications, handling permissions already granted or denied, and prompting the user for notification permissions. Bahmutov demonstrates writing and executing tests that simulate different conditions by manipulating the Notification API properties and using stubs and spies to verify the app's behavior without causing numerous popups during testing. The post emphasizes the importance of testing the web application code rather than the browser's implementation of the Notification API to ensure comprehensive coverage of the app's features.