I would appreciate it if you could explain precisely what it is, how it proves the earth is moving
It does not prove the earth is moving. It only proves something is moving relative to something else.
It's commonly known as The Doppler Effect. It's the basis of our advanced radar systems.
Listen next time an emergency vehicle approaches and you will hear the siren develop a higher pitch as it approaches. The distance between waves decreases because the vehicle moves in physical space. Less distance means higher frequency means higher pitch.
This works with all waves - light waves included.
In the flat earth model, the sun and moon would be moving toward you or away from you at all times except for one instance for each where it is right above you and you see the pure light. Both objects would have a violet hue as they approached you and a red hue as they moved away. We see the same yellow light at all times because our distance to the sun does not change much or fast. Same with the moon.
The stars are all emitting a red hue. They are all moving away from us, so they could not be on a dome surrounding the planet.
