Can anyone demystify this for me?
The form auto-fill behavior depends is part of the browser UI, and can be configured in its option menu.
The a "remember me" checkbox sets an identifying cookie with a late expiry date. Until that date, and unless you log out, the web site will recognize you (the server keeps a registry of what ID number correspond to which user). No need to authenticate on connection because the cookie is sent with each request.
Without the remember me option, the expiry time is short, say 30-60 minutes, but it may be renewed as long as the user is active. If you're inactive for a longer period, the cookie will be discarded, and the site will not recognize you anymore.
When you log out, the session reference is deleted on the server, and, optionally, the cookie is cleared in the browser.
This can be dictated by the website as to whether or not this is allowable.