> What everyday UI thing is better done with this than other methods?

All of them.

> Why is it better?

Because it specifies a workflow that actually matches what you want to do instead of employing a pile of ad-hoc rules that quickly become an unmaintainable mess.

If you have animations and async calls and multiple screens/pages, you have an application with application states which transition in response to inputs. That's what state machines are designed to handle.

Storyboards are state machines. Navigation components are state machines. UI binding layers like redux are state machines.