Yes, giving an application access to your storage will, in theory, allow it to access the files and folders on your device and SD card. However, if you're worried about credit card numbers, contact list, call logs and things like that, those are not included in the access level that the app gets.
Furthermore, Google is actually working as hard as it can to prevent applications from using their access level for spying on users and when you're dealing with popular names like Asphalt 8, these have been thoroughly vetted, so you shouldn't worry too much about it.
