If the speed of light is the same whether we more or not, according to Einstein, then why do we see a redshift if the light source is moving away at great speed?
You can think of it in terms of change in frequency.
Light moves (in vacuum) at the same velocity regardless of it being RED light or BLUE light (different frequencies).
Immagine you see 2 platoons of soldiers:
The first one is made up of tall soldiers bobbing up and down (their heads) all together while running with a certain velocity. You can "see" the frequency of the movement of their heads..up-down-up-down...etc.;
The second one is made up of small soldiers that to keep up with the first (same velocity) have to adapt and bob up and down more frequently!
Hope it is not confusing.