Like many other things pertaining to IOS 7, the step for deleting videos is different from previous versions. (For versions other than IOS 7 directions are here.) There are more steps involved, but in a way the process is simpler. These are the steps,
Settings > General > Usage > Storage > Videos > Edit > Delete Item
When you get to the last screen, you will see your movies, tv shows, and music videos listed and there will be a minus sign that you just click on.
When you get to the Storage screen you may not see Videos as one of the options. That is because you have a lot of apps on your device. Just scroll down to the last one listed and select “Show All Apps. Then you can select Videos and open the list.