Understanding a circuit for reading current passing through human bodies
The basic principle is this:

Here, from an engineering perspective, is a human - 



The value of a human is much easier to measure or estimate for this model. Hopes and dreams do not enter the equation, instead it's just the sweatiness of their skin. The value of a human can range from 1kΩ to 100kΩ. 

We have a sweaty human, they are 5kΩ.

When we add a human to the circuit you provided - 







simulate this circuit &ndash; Schematic created using CircuitLab

Our human creates a voltage divider with R1. The circuit designer modeled their average human resistor as 39kΩ. We can infer this because the maximum change in a voltage divider is when the resistors are equal. Thus, to maximize the change in the signal being measured they set R1 to 39kΩ. 

With no human the voltage on \$A0\$ will simply be equal to AREF and the capacitor will be charged to the same. When the human is added and the circuit is complete, the voltage is now:

$$ A0 = AREF * \frac{(Human + 1kΩ)\;\|\;1kΩ}{39kΩ + ((Human + 1kΩ)\;\|\;1kΩ)}  $$

The capacitor is there too, but if we consider it to have infinite DC resistance then it won't enter the equation. It's there to smooth out the change in voltage.

You may have not immediately recognized this as a voltage divider because of the series 1kΩ resistors. Those are there to provide isolation to the human and, along with the capacitor, as a debounce circuit. Additionally the AREF for the 'supply' voltage. We're making an analog measurement so the analog reference (AREF) voltage is a good choice. Very little current is required to charge the capacitor in sufficient time.