Staying logged in only means a cookie is put on your computer so that when you return to the website, your login is already known. It doesn't mean you're linked to the website when you're not on it.
I generally stay logged in to avoid having to type in my username and password every time I come here (or to some other board or website I go to regularly).
However, I also wipe my cookies and cache frequently, so I have to log back in after doing that.