This article describes how to use YouTube's Clips feature to quickly cut out your favorite parts from any YouTube video (livestreamed or not) and save or share them on a loop later.
How to Make and Share YouTube Clips
The clipping feature is currently available for Android and desktop users via the official YouTube app/website. It will come to iOS in the future.
Select Clip under the video. It may be in the overflow menu.
Make your clip selection. Do this by dragging the left and right edges of the blue box to expand or minimize the clip as desired. If you are using the desktop site, you can also adjust it manually by entering the time. Drag the entire box to change the clip location without changing the duration.