When you first logged in to Stack Overflow, a cookie was placed on your machine which remembers your login. Next time you go to any Stack Overflow related page, you will be recognized because of this cookie and show up as "logged in".
When you ask explicitly to be taken to the login page, that's where you are taken. The fact that you are already logged in doesn't matter. You might be wanting to change to a different user name.
Just to clarify some more: (disclaimer: I am deducing this from the described behavior, not because I have inside knowledge about the workings of the SO servers)
If you open a new browser session, but there is a cookie that shows you had been logged in previously and didn't log out, then you are considered "provisionally logged in". If you go to a particular question, you will be able to answer/edit/vote etc with the privileges of the last logged in person (usually, that's you). However, if you explicitly request the login page, you get taken there whether or not the cookie was set that indicated you had previously logged in and not out. This accounts for you arriving at a shared computer, and finding someone else had been using SO with a different user name.
Once you have established yourself as a logged in person, you are considered logged in for the duration of the session - that is, until you either log out explicitly, or close the browser. A logged in person doesn't need to be offered a login screen. Therefore, if you ask for the login screen you get redirected to the home page.
I hope this makes sense.