You can close Bunzilla by clicking on the "exit" button, selecting the exit option from the file menu, or closing the main window. By default, Bunzilla will continue to run in the background, sharing files, uploading and downloading the files you requested etc. To restore your session after closing Bunzilla, simply click on the program icon within the system tray area. If you wish to completely turn off Bunzilla, you should right click the icon in the system tray and pick exit.
You can modify this default behavior by unchecking the "Continue to run as system tray application on close" option on the General configuration tab.