The earliest Waveform Monitors were probably appropriately configured oscilloscopes, used in the development of early television. As such, their operating principle is easy to understand; in an old style cathode-ray-tube oscilloscope, as the beam moves horizontally across the display, once per video line, it is deflected vertically by the voltage of the video signal. Thus, the horizontal axis represents the horizontal axis of the image, and the vertical axis represents brightness.
The test image as viewed on a Waveform Monitor. Notice the peaks for the bright reflected light on the wet concrete, above the shadow detail for darker parts of the image
Most Waveform Monitors are capable of reading a single row of pixels, resulting in a display of a solid, unbroken trace across the display representing the luminance of that row. More usefully for the cinematographer, a Waveform Monitor can usually provide simultaneous analysis of every line in the image, scanning and compiling all of them into the same display with variable transparency.
A single line waveform trace. Notice the trace follows the brightness of the row of pixels in the image (indicated above in red)