Light behind is red-shifted. It would not be black, that is an error in this video, but rather you would begin to see UV and higher wavelengths that are now shifted into visible. And a spacecraft would presumably have multi-spectral cameras that can see in more than just the visible anyway.
And you'd assume that a species that has figured out how to travel at those speeds is able to show a display in 'false color' showing it as it should have been rather than the redshifted image. It's a pretty trivial correction if you have the sensor capability.
EDIT: Also the blue shift in the front is totally not what you would see either. That's assuming no EM radiation exists outside of the visible. What you'd actually see is infra-red becoming actually-red as lower energy wavelengths are blue-shifted into the visible.
