Stars help you advance in level. The star is filling in with every solved puzzle.
What level you are at shows the number next to the star. Currently the highest level in the app is 400.
You may tap on the Level Star to see how many stars remain until the next level.
For reaching new levels you receive rewards.
With every odd level achievement you receive coins and with every even level achievement you receive a free pack that can be downloaded from the Packs for stars category in the Puzzle Shop.
To find it please open the app, open the Puzzle Shop:
In the bottom left corner you'll see a Categories button (three horizontal lines). Tap it and scroll down to the Packs for Stars category.