the DGM thread covers this quite well
I like to think of it as a capacitor. If you have a light bulb and capacitor in parallel connected to a DC source, the bulb won't be lit right away. As the capacitor charges more current is routed through the bulb and it becomes brighter, until it reaches its maximum brightness as if there was no capacitor. If the power source is disconnected, the bulb gradually fades as the capacitor discharges its stored energy through the bulb. No energy is lost.
Of course there are many other real-life examples of this dynamic. Electric radiative space heaters come to mind - you turn them on and only after a while they become hot and heat the room. The dynamic is the same also quantitatively - in all 3 cases it can be modeled with (dx / dt) = a - bx, where a becomes 0 when the source is discontinued, and where x is itself the derivative of some quantity of interest.