Link an Apple iCloud mailbox to Mailboard AI by generating an app-specific password at Apple and pasting it with your iCloud email address.
Apple iCloud does not allow apps to sign in with your normal Apple ID password. Instead you create a one-off app-specific password at Apple and give that to Mailboard AI along with your iCloud email address. The Connect iCloud Mail screen guides you through it in three steps. The password is stored encrypted on your device and can be revoked from your Apple ID at any time.
appleid.apple.com → Sign In & Security → App-Specific Passwords. Generate one (label it e.g. "awaBerry EmailBoard"); Apple shows a 16-character password in the form xxxx-xxxx-xxxx-xxxx.you@icloud.com (or @me.com / @mac.com) address.