Redshift is caused by the motion of the object emitting the light and the fax that light behaves like a wave.

Consider the following example. Imagine a train sitting still on the tracks. The sound waves emitted by the engine propagate away from it as expanding, concentric spheres. There is no motion so an observer hears the sound at their natural frequency.

Then the train starts to move. At each interval, it emits a new spherical wave that has moved from the position where the last sphere was emitted. That means that the distance between the sphere just emitted and the previously emitted sphere in front of the train is closer than the distance between them in the back of the train. When the observer hears the sound, they hear a higher pitched sound as the train approaches them (blue shifted) and a lower pitched sound as it recedes from them (red shifted).

The same thing happens with the light emitted from stars in a galaxy. The color of the light changes because it’s light and not sound. But it’s all caused by the way we perceive light (or sound for the train) and not anything intrinsic about the universe.