You're implicitly assuming that what you asked the LLM to do is unrepresented in the training data. That assumption is usually faulty - very few of the ideas and concepts we come up with in our everyday lives are truly new.
All that being said, the refine.ink tool certainly has an interesting approach, which I'm not sure I've seen before. They review a single piece of writing, and it takes up to an hour, and it costs $50. They are probably running the LLM very painstakingly and repeatedly over combinations of sections of your text, allowing it to reason about the things you've written in a lot more detail than you get with a plain run of a long-context model (due to the limitations of sparse attention).
It's neat. I wonder about what other kinds of tasks we could improve AI performance at by scaling time and money (which, in the grand scheme, is usually still a bargain compared to a human worker).