The web storage only stores the values you put in to it - it doesn't store the game state automatically. You can make a save game by storing all the necessary information in WebStorage though. A really simple way would be to store the current level, for example. Then you can restore them to the last level rather than having to save every single object's position, angle etc.
WebStorage stores the data in the browser on the user's computer. So as long as you're using the same browser and visiting the same web page, the values are remembered.