A pulsating or variable star is a star whose apparent magnitude fluctuates. This change is due to the change in the luminosity of the stars. The changes in luminosity was then attributed to the swelling up and shrinking of these stars.
The luminosity is now understood as a function of the Pulsation period of the star.
The Question now was to find why these stars were pulsating. The answer was that it had to pulsates to keep the stellar structure stable. The star emits heat and light due to the nuclear reactions in the core. This heat however does not get fully dissipated and instead is partially used in the expansion of the stellar volume. This expanded gas now falls back to position thus dissipating the rest of the energy. This expansion and compression cycle is the reason for the cyclic changes in the luminosity of the star.