"hamburger menu" refers to a menu which pops out when you click on the 'menu' icon, which looks a bit like a stylised hamburger. => ☰

I agree that the article was not effective in introducing the topic and the problem space. I'm interested in mobile ui, but not familiar with all the new terminology, so this article was difficult to read. I scrolled down until I got to the conclusion, which was fortunately fairly clear, and then went back to understand the beginning.

It didn't help that the screen animations were too fast and lacked context. They also lacked user motions. Aren't there some standard ways of displaying tap and swipe events on a mobile screencast?

If you couldn't figure out what hamburger menus are after reading part of the article (presumes my including seeing the 3 images of such at the top) then you just weren't trying.

A hamburger menu is a single menu button (with horizontal lines on it) that folds out into a (much to) big list of menu items.