As a primer for "what does the capacitor do?"
A cap has the general effect of "opposing a change in voltage", so when we have one on the battery pigtail or the ESC power pads the effect is to 1) clean up (reduce) any transient voltage spikes 2) feed voltage back onto the line if there is a sudden drop. This can help keep voltage spikes that occur when we plug in the battery "smoothed out" and can feed extra power into the lines when we goose the throttle.
