Consider an analogy
Consider two water tanks with first tank having a higher water level in it. These tanks are connected to each other as shown in figure.
Case 1: The interconnect is a water filled hollow pipe: The water will flow from left tank to right tank.
Case 2: The interconnect is a hollow pipe filled with ice: There won't be any water flow.
In both cases, the pipes are filled with same molecules (\$H_2O\$). But the flow will happen only if the molecules are free to flow. Bounded molecules (as in ice), won't contribute to flow.
Electrons are like these water molecules. They will contribute to flow (current) only if they are mobile. So the excess electrons from negative plate to the positive plate will flow only if the interconnect have free (mobile) electrons in it.
Offer him some water (or other beverage) to drink.
What happens when you drink a lot?
You have to pee, which means there's some pressure that urges you to pee.
This pressure is
Usually people do not pee in suburban trains.
There's a resistance against the pressure.
On a toilet, human brains lower the resistance and there's a flow of pee.
This flow is
The water analogy is a very common one (I'm surprised you haven't heard about it) It usually explains
voltage as potential energy of water (from what height it's coming from), the
current as the amount of water flowing and the
resistance as the diameter of the tubing. (or something similar)
People go to great lengths with this analogy, finding substitutes for capacitors, inductors, etc.
You can find a lot of other analogies, because a lot of physics (our world) is driven by a difference in a certain value (different air pressure, different temperatures, ...) and some value changing, lowering that difference (wind blowing, heat exchange, ...).
In a metal like copper some of the electrons are not bound to individual atoms. If a voltage is applied across a copper wire these free electrons flow from one atom to the next. This flow of electrons is an electric current but the copper atoms themselves do not move so the copper wire does not vanish. Of course, the voltage source must supply additional electrons for the flow to continue.
As an analogy, think of a waterfall -- the electric current is like the water molecules falling, and the water molecules are falling due to gravity (which is analogous to voltage). The waterfall must be supplied with more water molecules for the waterfall to continue to exist, but the riverbed over which the water molecules flow (analogous to the copper atoms) do not move or vanish.