This is a port of the Draft extension for Chrome to Firefox.
You can install it directly from the repository.
Note: This addon doesn't work in private browsing mode.
Just press the button on the addon bar, and select new or previous to start editing on a new document or to open a previously opened one. There's also a button in the navigation bar.
If you are a keyboard guy, press Cmd/Ctrl + Shift + O to start editing in a new Draft document instantaneously.