Positive and negative assignments in electronics are purely by whatever convention a field/culture uses. Though physically electrons do flow from the negative terminal (electron flow convention), it is often simpler to do calculations the other way around ("conventional" flow notation / passive sign convention)
We use conventional in my circuits class, but electron flow in physics. Either way, you get the same result, but have to interpret the raw result differently.
I don't know the exact mechanism by which the arrangement works, but it probably deals with some form of electric force causing copper to dissolve in water, then redeposit on the rod. Probably electron stealing/giving is my guess, but I'm not a chemist.
The brownish foam is the copper fouling; follow the directions and keep going.