Inductors store energy in the form of a magnetic field. When current flows through the windings in the inductor, energy is stored in the magnetic field. When the amount of current coming in through the windings drops, the magnetic field collapses and produces current in the wires, hopefully long enough to maintain a constant level of current. In reality, the current produced by the collapse isn't enough to completely smooth out the pulsing. Capacitors store electric potential, much the same way as a rechargeable battery does.
Electrolytic capacitors have very high voltage capacity and high storage capacity.
Adding filter caps to the power circuit is easier and cheaper than adding more inductors. Filter caps usually come in two flavors. The caps in Marshalls actually contain two or more capacitors in one container. Filter caps can be added before or after the inductor in the circuit. This combination of using the center tapped transformer, the rectifiers, the inductor, and the filter caps results in a relatively constant power flow through the amplifier.