The Platinum Flag is simply a visual indicator that you have defeated all eight Gym Leaders and the Elite Four, signifying that you have completed the primary storyline of the game.
Here are some things that DO happen when you achieve Platinum Flag:
* You are able to challenge the Champion Cynthia.
* You gain access to the post-game content, such as the Battle Frontier, the ability to breed Pokémon, and various new areas to explore.
* You can continue your journey by collecting other badges and participating in other activities like contests and the Battle Tower.
So, while it doesn't offer any specific "rewards" in the traditional sense, achieving the Platinum Flag marks the completion of the main game and opens up a vast world of post-game activities.