I'm a casual GitHub user, and for me the fork button enabled me to grab a copy of some other repo, make changes , commit to my github repo without perturbing the main project and still keeping an online history at github. Its a step above just downloading the zip file and below getting commit rights to the original project. If I just made a personal branch, then I would have everything but the online backup because I couldn't commit.
One example is used by many online learning companies: they provide some baseline of code for use in a course. you need to get it to use in the lessons (edit: and make changes that you want to save). You can download a zip, clone or fork the repo. zip and clone don't get you the online backup.
It would be interesting to see how many people use Github for the reasons cited in the article (using a fork as staging for a pull request) or like I do.
As I wrote this, it made me realize I am a parasite on GitHub and the projects I fork, since I rarely contribute back (mainly because I don't have anything useful).