that came from an understanding of wave principles.
its the same for train whistles.
if you stand near a train crossing, as the train approaches, the whistle goes higher "redshifting" in to higher frequencies. this is because sound cant change its speed for a given temp and pressure. therefore the waves are closer together, increasing the frequency of the sound. after the train passes, the waves are farther apart causing the sound to drop (the same as blue-shifting for light). doppler radar does the same thing with radar to give a better understanding of clouds.
astronomers have seen that galaxies that are moving away from us have a distinctive redshift and those (few i believe) moving towards us, are blueshifted.
with the observer, the light cant move at any different speed that it does, therefore the light will be redshifted as he moves forward.
