The colour of light is determined by its frequency and wavelength. The longer the wavelength, the redder the light. As an object which is emitting a constant light moves away from you, the distance between the peaks and troughs of the wave of light are larger than if the same object were stationary, as the object is that much further away by the time it emits the next peak/trough. Think of it like a leaking water bottle. While stationary, the drops all fall near each other on the ground. Then, when you start running, the water drops land much farther apart. Imagine each of those water drops represents the crest of a wave. When the source of the water or wave is moving, the drops or crests of the wave are further apart. Bringing the metaphor back to waves, this means that the wavelength is longer, which, like I said earlier means the object is more red, or at least the colours of the light being emitted from it look like they are further down the red end of the spectrum. If the object is coming closer, blue shift occurs for the same reason, blue being of shorter wavelength than red.