After taking a quick look it seems the attiny is used to send a signal to the Pi to turn it off safely before removing the power with a relay, it basically removes the human component from switching the Pi off, this is done to avoid data corruption.
The capacitor only seems to remove the noise from the power supply of the attiny according to the circuit.
Capacitors are like a pond in a stream, the bigger the pond, the calmer the water. The same applies to electronics, a big capacitor will smooth all the ripples.
In this application anything avobe 100uF will do fine. Calculation of the capacitor value isn't really important in this case.