Digital Rotary Encoder -- Broken? Or am I misunderstanding how it works?
Amazon, like eBay, isn't the best place to get parts with proper documentation, as you have noticed. However it does say "with push-button", which, if true, means there's at least 2 pins that connect if, and only if, the shaft is pushed in. Leaves 3 pins, so that means there's one centre pin, which connects to an A and a B pin in shifted phase when you rotate. 

Centre connects to Pin A, then one step later the centre connects to pin B, then one step later the centre disconnects from pin A, then another step later it disconnects from B. Et cetera.

Most common layout I know of with that type is "centre pin"  is the middle one in the row fo three and B and A are on the other side. The pins surrounding the mid-pin actually being the push button. 

However, it may be entirely different. This would normally be listed in the Datasheet, which, as you have noticed when buying from "random sources" is a point of trouble more often than not.
Update: As it turns out from James' comment, the pins for the button are the two separate ones, but I was right on the middle pin.

When starting out with a new subject it's often wiser to buy only after you verify that there's an actual datasheet available, rather than save a dollar and then spend days not quite figuring it out. There's plenty sources in the places that Amazon caters to that make an attempt at supplying datasheets. While I'm not a super fan, sites like Sparkfun and Adafruit, for example, often have something of a datasheet for their parts. So do Digikey, Mouser and Farnell, of course, usually from the manufacturer, but you may need special accounts for those, depending on your location.