If you update your Android OS to version 6.x (or if you already have Android 6.x and download Angry Birds Action! for the first time), then Angry Birds Action! will prompt you to grant the game access to your photos, media and storage. It's only the 'storage' part that we're interested in.
The reason the game needs access to your device's storage specifically is because of application file size restrictions in the Google Play Store. These restrictions mean that instead of the more familiar method of downloading just one game file, Angry Birds Action! has to be sent to your device in two separate files.
This leads to Angry Birds Action! needing access to your device storage in order to load that second file which will allow the game to run. If you don't grant the game access, it will not be able to load.
If you have denied this permission by clicking "Don't ask me again", you have to manually allow the permission from your device settings to be able to play the game. You can do this by going to Settings -> Applications -> Application Manager ->AB Action! -> Permissions, and tapping the button next to "Storage" so that it changes its color to blue.
Please note that Angry Birds Action! will never access the personal files on your device. The game will only have access to the game files.