The Web App Manifest is the web technology that enables you to add your web app to a user's homescreen. This feature is commonly referred to as "Add to Homescreen (A2HS)".
For a hands-on, step-by-step guide on adding A2HS support in an existing application, check out the following codelab: Add Your Web App to a User's Home Screen.
For a more loosely-structured guide that goes into more depth about Web App Manifests, see Improve User Experiences with a Web App Manifest.
Use what you learn in these guides to add A2HS support in your own web app.
You can emulate and test A2HS events in Chrome DevTools. See the following section for more help: Web App Manifest.
Lighthouse fetches the manifest and verifies that it has data. The manifest that Lighthouse fetches is separate from the one that Chrome is using on the page, which can possibly cause inaccurate results.