You must check that box any time you have to "log on" to POF (that is, enter your ID and password) if you don't want to have to log on for future visits. Doing so causes OPF to (then) deposit a "cookie" on your computer - and entry in a special text file that, when you later access OPF (perhaps after your browser has been closed and then reopened), will let it recognize who you are and automatically log you in.
If in fact that happens (and the browser is set to retain the cookie), and you are automatically logged in when you return to OPF, you do not need to check the "Remember Me" box again
But if your browser is not set to retain cookies, then none of this will happen.
Is that what "being connected to" OPF means?
When you first access a site, a "connection" is established. That connection remains until you close your browser (or the tab pertaining to that "connection").
This is true even if you "log off" from the site. Then you are still connected to it in the communications sense, just no longer "recognized" and thus given certain privileges.
Best regards,
Doug