Open up the app, and go to "Account management". Tap the button and follow the instructions to confirm.
Publicly visible text/images often aren't properly deleted even when you succeed in deleting an account. Try editing or deleting them manually before deleting the account itself. If you're unsure what happens to your tracks: this can be found mostly in the Terms of Service/Privacy Policy; otherwise you can always contact Flipboard and ask personally. By the way, deletion requests don't necessarily mean your data will actually be deleted (e.g. due to legal obligations).
If you signed up for Flipboard using a third-party login such as Facebook or Twitter, you probably want to revoke all access from Flipboard. Click the icon below to find out how.