Seriously Cap! (pun intended) The storage they were referring to, no they don't. Capacitors are not batteries.
They're "not batteries" to the extent where they don't generate electricity. But they absolutely do "store" electricity, especially in power supply, and voltage regulator circuits.
The only thing that prevents them from storing electricity is the load from the connected circuitry. You really could take one of those big electrolytics in a PSU, charge it from the supply line, walk around the block with it, come back, and read very close to the same voltage you put into it.
The "filtering", and the "stabilizing", are both predicated on the storage function of the caps.
Now, once upon a time, in the dark ages before digital synthesis being used to tune to a given frequency, they used variable capacitors to tune radios and TVs.
Here's a variable tuner capacitor, from days of yore:
The capacity increases as the leaves mesh more.
So, a cap in league with a resistor or some form of inductor (choke coil), and you can tune a circuit to a given frequency or "band pass".
A third common function of capacitors in electronics is to block DC. A capacitor will pass AC as though it wasn't there (*), but will completely block DC from moving through it. These are use between stages of an amplifier, where you want the AC signal to pass to the next stage, but you want to maintain the supply voltage within the stage the signal is leaving. If you allowed the DC to pass between stages, all you'd accomplish is making the DC voltage higher at the transistor collector of the following stage.
If you think about it, the reason they call them capacitors, is because they have "capacity", which in this case, is a synonym for "storage capability". In that sense, they are "batteries" The storage and discharge simply isn't prolonged as would be the case with a lead acid type of storage battery.
(*) The reason for that is the AC wave charges one side of the cap, and then it releases it when the voltage shifts polarity. Then, the other side of the cap charges, and the cycle repeats.
So, here's a quote from Wiki: "The large capacitance of electrolytic capacitors makes them particularly suitable for passing or bypassing low-frequency signals up to some mega-hertz
and for storing large amounts of energy. They are widely used for decoupling or noise filtering in
power supplies and DC link circuits for
variable-frequency drives, for coupling signals between
amplifier stages, and storing energy as in a
flashlamp".
And the attendant link:
https://en.wikipedia.org/wiki/Electrolytic_capacitor