Checking for updates requires having an HTTP endpoint that reports the project's latest version available. You can also put additional information there, such as its release date, release notes, etc. For this, a JSON file is often used. You can host this JSON file in a GitHub repository and fetch its raw URL using the HTTPRequest node.
Look at this script for inspiration: https://github.com/Calinou/escape-space/blob/v1.2.0/menu/update_checker.gd
The above script uses the GitHub Releases API since the project is developed on GitHub, but this only works if the repository is public and releases are tagged accordingly.