In the fitness demo app, users can toggle an "incognito mode" that allows them to display their video feed only to the instructor, while keeping it hidden from other participants. This feature is implemented using a shared state mechanism provided by the Daily platform, which enables the app to persistently share data among call participants. The implementation involves adding a toggle input in the Header component for students to turn on incognito mode, and updating the visible participants list in the GridView component based on the local participant's incognito status. Additionally, the speaker view layout needs to be updated to filter out remote participants when incognito mode is enabled. Overall, this feature aims to provide users with a sense of privacy and security during online fitness classes.