Good question: There are at least three uses I can see for a magic eye tube monitor.
1. With a mono source feeding both channels of your system and the monitor connected to the speaker outputs, you can check and adjust electrical balance of the left and right channels.
2. With the monitor connected to the speakers, you can set the sensitivity control to a a peak power setting that will just close the eye. Thus, you can use it as a peak power monitor.
3. Connected anywhere in the system, it gives a visual indication of signal presence and, especially with the low speed (averaging) setting, it offers a visually interesting display, especially in a darkened room.
Lloyd