But after you watch them, how do you delete the video to free up space?
It's not so obvious.
Here's a set of old instructions on how to do it for the Kindle Fire. I believe the Kindle Fire is the same (or very similar) to the cheapie Fire tablet that I have. If so, the instructions are a little different now.
Here's how I did it (mostly following the instructions, at first anyway).
1) tap Video
2) tap Library
3) tap the three horizontal lines at the top left
4) tap Downloads
5) tap and hold the video you want to delete (in my case it's a TV show)
6) tap View Season Details
7) scroll up to see the episode list, the videos that have been downloaded will have a trash can icon on the right
8) tap the trash can icon and you will get the message "Do you want to remove this video from your device?"
9) tap OK to delete the video