In order to bring up the ability to screen record you can either click on the "Screen capture" button on the shelf quick settings or press the Ctrl, Shift, and Overview keys. You will then see floating bar at the bottom of your Chromebook's screen.
You will then see the ability to take a screen shot, either full or partial. You can then find the screenshot in the files of your Chromebook and choose images.
But for the fun part is the new screen recorder. Once you select the screen recorder you have the same option to record either a certain section of your screen or the whole screen.
There is a 3 second countdown and then the recording starts. To end the recording you simple need to select the option to stop recording at the bottom right of the toolbar. The video is now stored in the video section of your files.
Embedded below is a 3 minute and 9 second YouTube tutorial.