What is a stiffening capacitor?
A stiffening capacitor is an extremely large value electrolytic capacitor. Capacitors have the ability (capacity) to store a charge on their "plates". The larger the capacitance, measured in units of Farad, the greater the charge a capacitor can store for a given voltage. Most automotive capacitors can store a charge at 16 to 20 volts, and are rated from about .5 Farad up to 2 Farad or even more. Capacitors have an extremely low internal resistance to current flow, much lower than that of a battery. Capacitors can therefore charge and discharge power much faster than a battery can.
How does it work?
All capacitors consist of two "plates", separated by an insulator called the "dielectric". Each plate has an external connector. Connected to a source of voltage, electrons are forced onto one plate, and off of the other plate. When the capacitor is fully charged, no more electrons will flow in or out. Disconnected from the source of voltage, the charge will remain between the plates, and the voltage can be measured between the capacitors terminals with a volt meter. A capacitor can store a charge for a very long time.
The capacitor is connected to the positive power line to the amplifier, and then grounded to the vehicle chassis. When initially connected, the capacitor will very quickly charge up to the voltage of the vehicle's battery, usually a little over 12V.