The light clock moves uniformly in the horizontal direction relative to us. We imagine we can see the light pulse. Since the clock starts on the left and ends up on the right, the path of the light pulse is involves horizontal movement, as well as the up and down part.

It is a postulate of special relativity that the speed of light (in a vacuum) is always measured to be the same. Since the pulse is travelling a greater total distance than that of a stationary clock the tick must take longer. This means the clock runs slow, since the ticks are too long as measured by stationary clocks.