Begin is an AWS development tool that helps developers ship web apps and APIs on the platform with ease, as demonstrated by Paul Chin, Jr., who updated his 5-year-old Ghostbusters site to a functional FASTstack application using Begin. To handle form submissions, Paul used Courier, a notification service provided by Begin, which allows for easy setup of contact forms, serverless functions, and integration with Slack applications. The process involves creating a new Begin application, adding an HTML form, handling the submit using a serverless function, setting up a new Slack application to send direct messages, and connecting it to the Courier notification. This stream showcases what's possible when building apps with Begin and Courier, making it easier to create on AWS.