Your browser runs on our server. Like any browser, it keeps cookies, logins, history and site data in a profile folder that belongs only to you. That is how you stay signed in to sites between sessions.
The folder is named by a one-way hash of your App Store subscription. We never see your name, email or Apple ID.
The picture of your browser travels to your phone over an encrypted connection. It is not recorded.
In the app, open Settings and tap Delete browser data. Your profile is erased at once. If you don't open Browser Bridge for 60 days, your profile is erased.
Payments are handled by Apple. We only receive Apple's signed proof that your subscription is active.
Questions: the support link on the App Store page.