What does 'capacitor jumping up and down' mean and what work does it do?
Forget about it.  Move on.  The author of that web site seems to be struggling with what a capacitor is himself.  He has formed little mental cruxes in a attempt to demystify these capacitator thingies to himself, much like early people created various myths to explain things they didn't understand either.  He then tries to explain the mysterious beast to you using his personal myths.  It doesn't work well.  Like I said, forget about it and move on.

I think his vision of "jumping around" is really referring to the common mode voltage, such as when used to pass a signal, which is different to him than when used for power supply smoothing.  Don't get hung up on this guy's personal mythology.