Electronic – Why does the voltage of a zener diode match the voltage at Vout

Look at the Zener diode curve. You will see that the device breaks down at the Zener voltage when reverse-biased, and conducts. That property will fix the output voltage at the breakdown voltage, over a range of output currents, when used with a resistor, with relatively small voltage changes. It will also stabilise the output against changes in the input voltage.

Strictly speaking, Zener diodes are low-voltage devices (up to about 5V6). Higher-voltage ones have a different mode of operation and are called avalanche diodes. Both types are commonly referred to as Zeners, though.

The reverse voltage of a zener diode does not show a sharp knee with increasing current, so the zener will conduct several milliamperes at voltages well below its rated voltage. There's no need for a fuse in this case assuming that the input impedance of your microcontroller is high. By the way, what is a "GHI Cobra"?

I suggest replacing the fuse with a 4.7k resistor and using a 5.1V or 5.6V zener. It would be best if you could reduce the normal operating range of the ADC input so that the maximum expected voltage is around 4V...perhaps by changing the 250 ohm resistor to 200 ohm. Then the soft knee of the zener will not affect your measurements and you are giving the zener a little room to start conducting.