What is the purpose of the masks? I'm confused.

If you put them on and then refresh at Shadowglen, you'll get infected with a zombie disease C:

Okay, so I have the disease.

Now, how do the stages advance? Like, how do you get from stage 1 of the disease to stage 2?

You just wait. (:

visit this thread! for any more questions you have regarding zombies. :)