It is best to start understanding the theorem for a particle in one-dimension, so that the phase space is 2-dimensional and Louiville's THM says that Hamiltonian evolution preserves areas in phase space. Since an initial condition is just a point, an area in phase space corresponds to a set of initial conditions. Each of these initial conditions becomes a trajectory, and at some later timethe points will all have moved somewhere else but they will still occupy the same total volume in phase space.
For contrast, take a damped pendulum, which is typically not a Hamiltonian system. Since the pendulum bob eventually comes to rest at a single point in phase space, no matter how large your initial area is it will shrink until it is only the size of one point.
