When the game is over, if no slots are claimed, you have a limited time to claim a slot before a timer expires before you can no longer claim scores for the leaderboard. If a player slot is selected during the game or immediately after, the session ends and the player is encouraged to submit their score.
The default behavior to to skip the prompt and present the Share Screen, but if you have elected to prompt before saving in your Playing Preferences, you will be shown your rank and your score's position on the leaderboard:
Save to Leaderboard: Only Appears If Set in Playing Preferences!
If you choose to throw the score away, you'll return to Waiting to Start. If you save, or have the Playing Preferences set to default, it will advance to the Sharing Screen. Note that submissions to the leaderboard are automatically reflected on anyone viewing that leaderboard, any online visualizations of the leaderboard, or any third party integrations that reference that leaderboard. The leaderboard referenced here is specific to that machine, not the global leaderboard.
If you don't see this screen, DON'T PANIC! This is normal, default behavior and you are often taken to the next screen, Sharing and Challenges.