When you close the SpiderOak application window, typically done by pressing an "X" or similar control, the application minimizes to run in the background, in keeping with its purpose. In order to completely quit the application and log out, right click on the SpiderOak icon in your computer's system tray or notification area and choose Quit. That will shut down the software and close your connection with our servers.
Before you close it down you might wish to enable the option "Ask for password at startup", which is found at Preferences - General. Otherwise anyone can open it without the password.
Do keep in mind that being a backup tool, SpiderOak copies files on your local computer; it does not remove them. Logging out will keep nosy people out of your SpiderOak account, but keeping the files on your computer safe from snoopers is your responsibility.