Unexpected Apple ID password prompt - Which App is causing this?
Perhaps this snippet from this help page at Appigo will provide a clue:


  iCloud is different from the other synchronization services supported by Todo in that Todo does not actively control when synchronization takes place. When you make changes to your tasks Todo flags its data as having changed; iCloud watches for this flag and synchronizes as needed. In short, while Todo can inform iCloud that it needs to be synchronized, it cannot start or control an active synchronization.


I think that ToDo needs you to manually login to iCloud, which uses your Apple ID for authentication, to sync because the saved password is incorrect. Perhaps you need to reconfirm the login credentials within the ToDo app so it syncs correctly.