I think you need to download the Amazon app, and I am sure there is a specific one for videos. Then you will watched the video, streaming, through the app. It doesn't directly download to your computer...thus saving space by not taking up room on your hard drive. However, to stream video/audio, you need to be connected to wifi.
Does this Chromebook have some sort of google store, or app store, where you go to download apps? The Amazon apps are all free downloads.