Simply put, the light pulse traces out different paths for each of the two observers. The one riding along with the clock sees it simply move straight up and down, while the observer on the sidelines watching the apparatus moving past him sees it trace out a diagonal, and therefore *longer*, path. Since light travels at the same speed for every observer, the light pulse MUST then take a longer time to travel the longer (diagonal) path (as seen from the point of view of the observer on the sidelines). Or "moving clocks tick more slowly than stationary ones".
What if the speed of light is not a constant?