A spectral line from the left side of Saturn's rings, as you see them, is at a wavelength slightly shorter than the same spectral line measured from the right side of Saturn’s rings. Which way (direction) is Saturn rotating, and why?
The light from the left side of the rings is slightly shorter (heading towards us) than the right side (heading away). So the rings are rotating from left to right.
Let's first talk about the Doppler Effect. It states that waves, whether sound or electromagnetic, shift depending on whether the thing producing the wave is approaching or receding from us. And if you listen for something that approaches or recedes at a high speed (like a car or a train), you'll hear the pitch be higher as it approaches and then lower as it passes us.
In sound, a higher pitch is made of sound waves that are shorter. A lower pitch is made of sound waves that are longer. And if you think about it, it makes sense: something that is approaching will be "crunching" the waves together, making them shorter, whereas something headed away from us will "drag" the sound waves and make them lower.
The same is true for light - things that are headed towards us will "crunch" the light, making the light waves shorter while things headed away from us will "drag" the light, making the waves longer.
So which electromagnetic waves are longer and which are shorter? One way to remember: one is to remember that almost everything in the Universe is expanding away from the Earth and so we say that they are red-shifted. Red-shift means heading away from Earth and blue-shift means heading towards Earth.
Ok - so let's look at the question about Saturn's rings: the light from the left side of the rings is slightly shorter (heading towards us) than the right side (heading away). So the rings are rotating from left to right.