Also, relativistic motion doesn't appear to affect the movements of the other actors, though it's kind of hard to tell for certain.
As you move in relativistic speeds, your eye hits more photos, even those going sideways (or backwards), because the photons at an angle are slower than your eye (imagine a car moving fast through the rain, even drops that go in the other direction will hit your windshield).
Therefore, your field of vision grows (you can see things behind you), but only when walking forward.
I haven't read anything about going backwards, but I imagine that your field of vision would shrink, as you are now faster than photons that would previously hit your eye. My guess is that you would only see photons coming directly to you from your front, until you reached the speed of light and went blind (or could see 360 degrees, moving forward).
The observable change is the separation between photons (wavelength) caused by a moving source.
(correct me if I'm wrong, I'm terrible at physics!)
It actually helps to just forget about relativity for a moment, and think about the angles light from distant objects will have to take to intercept you. Then it becomes more clear.
I'm not sure I'm missing a step going from "wider fov" to "things seem farther away", maybe I am...
It would be nice to see a write up of exactly what effects they considered, though.
One is that the actual size of things changes. That's length contraction.
But the other is that, because you are moving at such high speeds, how you observe objects is also affected. That's what StavrosK discusses in the adjacent post.