This circuit is a very simple logic analyzer. You probe something, and the circuit will tell you if the probed point is a logical HI or LO by lighting one of the two LEDs. Because an inverter chip has its own threshold before it output an inverted signal, it can also be used as a simple analog to digital converter. The surrounding resistors are used to specify the exact threshold you want to use as a trigger.
Another function of this type of circuit is to ensure one of the LEDs only lights when there is an input at the probe.
If the input is Logic HI, current flows probed point to ground, causing the bottom NOT gate to output a LO. Current flows from the source through the green LED with the bottom NOT gate serving as a sink. The top NOT gate outputs a LO, so the red LED is off.
If the input is Logic LO, current flows from the source to the probed point, with the top NOT gate inverting the signal to output a HI. The top NOT gate then serves as a current source for the red LED. The bottom NOT gate outputs a HI, so the green LED is off.
If there is no input, the top NOT gate is pulled HI with a LO output, and the bottom NOT gate is pulled LO with a HI output, so neither LED will light.
I built a very simple digital design assistant a while ago with a similar circuit, although mine has less functionality. I used a single inverter to do essentially the same thing.
If the input is HI, one LED will light. If the input is LO, the other LED will light. The inverter serves as either the source or the sink, depending upon the input. If there is no input, neither LED will light. The difference in this circuit from the one you showed is that mine does not have the resistors to set the exact desired threshold for triggering - it relies on the chip's internal threshold to function.
Typically, this type of switching would be performed by complimentary mosfet transistors, as noted in the original circuit description. The shown circuit just shows how to perform a similar function with a commonly acquired logic gate.