You can't really save your notes unless you sign up. Mind you, using this as a note-taking app does not require you to sign up. This is only if you want to save your progress/notes.
The sign up process is easy. Though Draft was made for quick note-taking, the company behind the product works really hard on team-work; which is why they give you the option to sign up with a company email (firstname.lastname@example.org) so that other teammates can have access to this (again, this is not required).
Regardless of which email you sign up with, the screen above is what you'll get. A personal dashboard where you can see all the notes you've created, your last changes, and you're given the option to share x note with others.
Once you sign up, the next time you open up a new tab to use Draft, it will show you the option to "Save to your private docs".
So to summarise: Draft is a Google Chrome or Brave extension that lets you open up a new tab and use it as a note-taking app (with a lot of added functionality). It's completely free to use, but you must sign up to save your notes. Enjoy!