This circuit is fairly simple.
First, most Oscilloscopes have an input impedance of 1 M\$\Omega\$. This is implemented in the above circuit by the resistor divider composed of R2 and R3 (assuming the TL064 has a infinite input impedance).
This divider also divides the voltage by \$\frac{R3}{R3+R2}\$, which in this case is equal to 0.12.
Second, the signal is fed through a non-inverting amplifier, with some small amount of gain (equivalent to \$1+\frac{R7}{R8}\$), or 1.111....
Finally, and offset is injected into the signal through the voltage divider composed of R1 and R4, and C2 serves to remove any high-frequency noise.
The actual design work is fairly simple.
In this case, the ADC has a reference of 2.048 volts. Therefore, you need to bias your signal so 0V input results in 1/2 full scale (which is 2.048V), or 1.024 volts at the ADC input. This is the motivation behind the R1/R4 voltage divider.
The input of the ADC is of unknown impedance, and presenting a known, fixed impedance to whatever you are connected to is a desirable thing to do. This is the motivation for the input resistor divider.
Lastly, you have to set all your values so that when the input voltage is at the maximum voltage you are interested, you are at or close to full scale on the ADCs. This can be done by adjusting the Op-Amp gain, or the input voltage divider (or both together).
The circuit is certainly bare bones. The TL064 is not a great Op-Amp, and if you pull the input voltage too low, the output of the voltage R1/R4 divider will go below 0V, and you will begin to pull current through the ATMEGA's integrated protection diodes, which is generally a bad idea.
There should be some Schottky diodes to clamp the input to the ADC if it deviates beyond the supply voltage range (either positive or negative).
It shouldn't have too much trouble if the input goes too high, since the output of the TL064 can only swing to within ~1.5V of the power rails, which would prevent it from getting too high (since 5 - 1.5 = 3.5, and 3.5 / 2 = 1.75).
On the other hand, I'm not sure if the op-amp can actually swing high enough to actually use the whole ADC range.
It sounds like you are inadvertently shorting out the resistor you hope to measure current through because the scope probes have a common earth point. If you have one scope probe's earth crocodile clip on the bottom of the lower resistor you can't arbitrarily connect the other probe's earth clip to a different point in the circuit because it shorts the two nodes together.
Take a step back and think what you are trying to do. If you connect one resistor (10 ohm) to your triboelectric generator and measure the voltage across this one resistor you can perfectly assume that the current through that resistor look exactly like the voltage across it.
This is basic ohms law i.e. I = V/R - if R is a constant then I is proportional to V and will be identical to it other than scale.
If it so happens that you see no voltage across the ten ohm resistor this is because your scope isn't sensitive enough. In which case, increase the resistor value or use a scope preamp.
Best Answer
Put the ground clips of two scope probes on the ground of the circuit (the one denoted with ground symbol). Then put Channel A on point A, channel B on point B, and then use the Math function of the scope to display the difference between two signals.