This is due to authentication issues with Game Center, but it is easily fixed by following a few steps.

  1. Open your Settings app and select General > Date & Time > Set Automatically. If it is already switched to "Set Automatically" toggle it off and manually set your correct time zone before continuing with the steps. You can switch it back to "Set Automatically" after the Scores are restored.
  2. Remain in Settings and scroll down to select 'Game Center' - on the following screen press the tab next to "Game Center" to log out, and then log back in.
  3. Force quit the affected app by pressing the home button twice and swiping upwards with your finger on the app.
  4. Reopen the app and try accessing your Scores again.

This video shows you how to log out of Game Center from the Settings app:

This Apple Support article demonstrates how to force close an app: