Why does this Edge Trigger Work on input of 555 monostable config?
The explanation of the capacitor on the  input is as follows. With the switch open the capacitor gets to a point of zero charge across itself. Then when the switch is initially pressed the switch side of the capacitor is suddenly taken to GND. Since the voltage across a capacitor cannot be changed instantly the initial switch to GND is also seen on the other side of the capacitor going into the 555 pin 2. Now as the switch is held closed the 555 side of the capacitor quickly starts to charge up toward the Vcc rail via the 10K resistor.

Note that when the switch is released the now charged capacitor will push the 555 pin 2 up to a voltage above Vcc for a period of time.

This is a simulation circuit to show how it works: