And though I don't fully understand all of this, I'm working on wrapping my head around the main concepts.
The easiest way to see this is:
Go for a stroll during a light drizzle with no wind and look over your left shoulder. You will see the water droplets falling directly downwards, and you will correctly perceive that they are coming from the cloud directly above you.
Now start walking and look over your left shoulder. The water droplets seem to come from someplace in front of you, falling at about 45 degrees. You could incorrectly deduce that they come from a cloud in front of you, not above.
Now picture yourself doing the same experiment with a star above your head. The water is replaced with light, but everything else stays the same. Instead of walking, it is the Earth that is moving around the Sun. You will have to repeat the experiment in winter and spring, for example, to see the effect of moving and not moving in some specified direction.