As far as I'm aware, Games like Angry birds etc, either save your progress locally (so if you wipe your fone, you have to start again from the beginning) or they store it on their own servers somewhere on the internet. Usually these kind of games require a log in of some kind (Usually FB or G+).
This is how I am trying to do it. Essentially I'm using a set of credentials generated from the users FB ID as a login to a MySQL DB where I am storing the users Hi Scores and level progress etc. I am using PHP scripts stored on the server to enter/retrieve values from the database using the HTTP commands.
Using dropbox to store this stuff limits your userbase to those who use Dropbox (I'm sure I wouldn't create an account just so I could play a freebie game I downloaded from the Play store), where as your own online database allows anyone with a net connection to play.
Hope that helps.
@CJB:
That's a very handy thing to know.