Potentiometer Ripple under load
As you turn the pot, the resistance it provides is effectively lowered. This draws more current from the capacitors, lowering their voltage more quickly. When the resistance is low enough, the capacitors aren't able to maintain a steady voltage between each AC cycle.

Another way to view it is this: the capacitors and the resistors make an RC low-pass filter. When the resistor is big, this filter has a long time constant. When the resistor is small, the filter has a short time constant. As the time constant becomes shorter, this means the cutoff frequency increases, and the filter attenuates the incoming AC less.

To reduce the ripple, you can decrease the load current by increasing the resistance, or you can add more capacitance.