Hi Daphne
It's nothing to worry about. You are logging out successfully but your computer has stored your username and password and is logging in for you when you come back. You may have clicked a button saying 'remember me' or similar on your web browser.
If you don't want your password to be stored you should delete your cookies.
In Internet Explorer, try going to Tools (on the menu) → Options → "Delete Cookies" (under the "General" section) and confirm to delete cookies.
But either way, your account is perfectly safe!
Kind regards
Olly, your editor