This is a tool that allows you to isolate UI components. The practical benefit is that you can list all of the expected states of a component without having to build the world around it.
I am currently working on displaying data that has no simple way of recreating accurately in environments below production. By using Storybook I can demonstrate what it would look like.
Another benefit is that you can test the component in isolation on a range of devices/browsers. That makes checking a new component much easier.