Netlify Forms is a way to process form submissions without the need to configure a backend, using Netlify's build bots that parse your markup at deploy time and create necessary form handlers and APIs. To integrate Netlify forms into a Vue app, you can add a netlify attribute to the form element, or use a pre-rendered static HTML file in the public folder, or utilize Vue Router with two components for submission success and failure views. The process involves adding form elements, updating data attributes, handling form submissions, and configuring router routes. Netlify Forms provides a way to collect valuable data without the hassle of configuring a backend, making it easier to integrate forms into favorite frameworks.