There are two ways to run draw.io offline.
Use the desktop application
You can download it from the official Github site. There are versions available for Window, Mac and Linux.
The desktop version is completely sandboxed, apart from a call to Github on startup to check for updates.
Running the Web Application as a Progressive Web Application (PWA)
The default version at app.diagrams.net runs as a PWA. It will download the application in the background, and allow you to work offline once the files are downloaded. Remember that you must visit this URL prior to going offline at least once in order to cache the application in your browser.