One of the main reasons of failure to authenticate with Google Drive is because you’ve previously allowed the app to have full access to Google Drive and are trying to set the app again. Allowing full access to Google Drive is not a good security practice and Google discourages this.
To fix this, you’ll need to remove the app’s existing connection from your Google account and then attempt to authenticate again, ensuring that you only allow the app to access its own files (limited access).

To remove the app’s existing connection, open your Google account’s Third-party apps and services page. On that screen search for “sms” and then select SMS Backup & Restore / Pro.

Next, select the option to “Delete all connections” and then confirm. The prompt will confirm that your existing data will NOT be deleted.

After the connection has been deleted, you can go back to our app on the phone and then try to log in again, selecting the option to only allow it to access its own files and the authentication should work this time.