When you Sign Out, you have signed out of the Lightroom installation on that computer, and so you can then Sign In to the Lightroom installation on another computer.
Note: When you attempt to sign out or sign in, you must have internet connection temporarily to allow the sign in/sign out.
But every time you sign out then sign back in you will need to re-enter the serial number. I think Adobe has a limit, a Time Limit, for signing out and back in.
If you continually switch from computer to computer you will run out of activations. IIRC the time limit is 48 hours but I'm not sure on that number. So if you Sign in on one computer then sign in on another computer shortly after that, 2 activation limit, if you then sign out on one of the computers you have to wait 48 hours before you can sign in on another computer. Why not buy 2 copies then you can install it on 4 computer and never have to jump through any hoops.
Also those 2 computers are not supposed to be used at the same time.
You don't sign out in the preferences, that is for Lightroom Mobile IIRC.
You go to the Help menu and Sign Out from there.