What's a "static web server"? I haven't seen this term. A webserver for only static pages, i.e. it won't talk to a runtime? But it's "for serving web apps", this means all the dynamic content is pulled from client side JS? Then you would still need programs on the server to answer AJAX requests. I did not find an explanation on the page or by Googling.
The trick is that most of the dynamic stuff you get from REST services on OTHER servers.
For example, people maintain static blogs with Jekyll and can enable commenting with an HTML widget.
Or your app can use MongoLab for it's back end data which again is a separate server.
It turns out you can do almost anything just developing with your own static files.
This is one of the reasons hosting prototype web apps on DropBox is popular.