When a capacitor is turned on, the voltage is stabilized to the source's voltage: I can understand a scenario where the voltage of a capacitor and the voltage source do not match in voltage. But it doesn't make sense for an inductor and the source's current to not match in current. They have to match in current because they are in series.
Voltage Stabilization: These capacitors act as local energy reservoirs, providing instantaneous current to ICs (integrated circuits) and other active components during transient conditions, thereby stabilizing the voltage levels.
Frequently capacitors are put across the power supply to hold the voltage steady. This works because the more capacitance you have, the harder it is to change the voltage, because it requires more current to do so. In this application, capacitors don't smooth energy, they smooth voltage.
As the capacitor charges up, the current gradually decreases until it reaches zero. Once the capacitor is fully charged, it stops accepting current, and the voltage across the capacitor remains constant. If the voltage across the capacitor is changed, the capacitor will either charge or discharge until it reaches the new voltage.
A Capacitor merely stores Energy. The energy will be returned to the circuit when the driving voltage drops. There has to be a resistive component in a circuit for Energy to be dissipated. So you either rely on the source resistance of the supply or you have to insert your own in the form of a series resistance.
The charge that a capacitor can store is proportional to the voltage across its plates. When a voltage is applied across the capacitor, the current flows from the voltage source to the capacitor plates. As the capacitor charges up, the current gradually decreases until it reaches zero.