how do I install the audacity app on my Apple Watch?

You don't. The Audacity App does not appear to have an app version for Apple Watch. The App is available and compatible for the following devices:

Compatibility

iPhone: Requires iOS 11.4 or later.

iPad: Requires iPadOS 11.4 or later.

iPod touch: Requires iOS 11.4 or later.

Mac: Requires macOS 11.0 or later and a Mac with Apple M1 chip.

iTunes, which hosts the Audacity app, says otherwise. Since Audacity is however, not an Apple product, contact Audacity directly for support with their product. Apple only hosts their app. Apple doesn't have anything to do with which devices their app runs on. That's entirely up to Audacity.