Consider the following light clock (discussed in the next section of your link):
In part (a), the entire apparatus is stationary, and the light simply reflects back and forth. In part (b), the entire apparatus moves. From the perspective of the apparatus, the light still reflects back and forth; but from the perspective of an observer watching the apparatus moves, it looks like the light is travelling at an angle.
Furthermore, as Feynman mentions in the first section, one of the consequences of Maxwell's equations is that the speed of light is independent of the motion of the source. So, there will be a difference in time taken when the entire apparatus is moving, since the light has to cover a longer distance (from the perspective of the observer) at the same velocity $c$.