Why does it ask for an emailed code instead of a password?

That's how Supernote's own sign-in works when you log in from a new place: it sends a one-time verification code to your email to confirm it's really you. The app simply shows you Supernote's login page, so you complete it the same way you would on Supernote's website. This way your account stays protected — your password is never used or seen, and to log in someone would need access to your email account.