The Logitech Control Center installs kernel extensions that intervene in the system and modify input on the kernel level.
Since Spacious does not access the hardware directly, but instead works on a higher, safer abstraction level, it can only pick up modified signals when the Logitech Control Center is installed. Accessing the original signals is no longer possible in that case, so that Spacious can not function as designed.
We therefore recommend to uninstall the Logitech Control Center, if you experience problems. You can uninstall the Logitech Control Center by executing the app "LCC Uninstaller.app" in /Applications/Utilities.