After watching a video, it will appear in your Watch History, which is accessible on the extended sidebar.
Click the button with three vertical lines at the top left corner of the screen to expand it. After it's expanded, Watch History will be the 7th button.
For the time being, Watch History isn't shared between devices, meaning videos watched on your computer won't appear in the watch history on your phone and vice versa.
You can delete specific videos from your history by hovering over them and clicking the "X" on the right. Or, you can delete your entire history by clicking Clear History.