Log Out vs Remember Me

Howdy,
I clicked the rememberer me when I log in, and evidently I really do not understand what that means. I had hoped that at least it would remember my user name for the next time that I log in. However, when I log out, I see a message about cookies being removed and one other thing that I can't recall, and when I come back to the site, I have to enter all the login info all over.
Have I missed something?
I am using an iMac on OS 10.9.5

Thank you for your help,
DN

In short in general if you tick the �remember me� box your username and password will be saved in a cookie on the computer you are using. This means that next time you log on you will not need to re-enter these details. If you delete the cookie or your browser is set to not accept cookies this function will not work.* You can de-select the function at any time by removing the tick from the box.

Note: we suggest you do not select the �remember me� function if you use a shared computer or non-secure access.

I don't know how our forum makes use of cookie, as cookies serve many functions, their most common purpose is to store user preferences, our admins or moderators might answer you more on this.