Categories in Youtube Gallery are made to group video lists.
To show all videos from Video List of selected Category you need to create a new Video List with the following:
videolist:catid
- Will insert all videos from Video Lists of Category with ID #4.
or
videolist:category:music
- Will insert all videos from Video Lists of Category "music"
To show list of categories on Front-end you need special plugin: /youtubegallery-category-list