A multistage rocket is, like any rocket, propelled by the recoil pressure of the burning gases it emits as it burns fuel. What characterizes it as "multistage" is that it successively jettisons one or more fuel cylinders as they become empty.
Essentially, the principle is that once the fuel in a section of the rocket cylinder is expended, that section becomes dead weight. By jettisoning the dead weight, less total fuel is needed to reach a given velocity. A big rocket can be used to push a smaller rocket into space. Then the big rocket is jettisoned and the small rocket can accelerate, starting from the speed it attained under the propulsion of the big rocket. This can, in principle, be done any number of times.
This concept was developed independently by at least three individuals: the Russian Konstantin Tsiolkovsky and the American Robert Goddard, and the ethnically German, Transylvanian-born Hermann Oberth.