The stationary light clock animation shows a pulse of light bouncing between the mirrors of a light clock. The pulse leaves a trail which becomes useful when a moving light clock is considered.
The moving light clock animation shows that the light pulse travels further. Since the speed of light is constant (a postulate) the clock must have a longer tick, and hence run slow. This can be quantified to give the time dilation formula.
The remaining animation compares a moving light clock with a chain of synchronized stationary light clocks.