Can anyone explain what is happening as you get a few nines in your velocity?
Yes, that's the microwave background being blueshifted, but why is it growing across the screen? Shouldn't the angle that's blueshifted be less and less as you go faster? At lower speeds we do see the starbow effect, passing stars going down the spectrum as the angle increases.
And what becomes of the stars as you're going fast enough? Yes, most of the energy is shifted out of view but don't stars emit some at such low energies?
These are good questions, but the answer is essentially "not in scope for this visualization". And, also, really hard to visualize with reasonably correct physics at higher "nines".
The tool does not deal with gravitation. We know that our local patch of the Milky Way (out to a few hundred lightyears) is full of kHz-Hz waves (LIGO, Virgo et al.), and nHz waves (pulsar timing arrays). Known sources at these frequencies are extragalactic, so there will be lots of these waves outside the Milky Way too. At significant boosts, these gravitational waves will impose visual distortions. The "twinkling" that pulsar timing arrays detect over the course of many months would be detectable over the course of much shorter periods of ship time.
Gravitational microlensing is likely to create uncomfortably hot spots (low-energy caustics from our usual point of view) for an ultraboosted spaceship.
We also strongly suspect there are lots and lots of black holes massive enough to impose a significant gravitational redshift on background light from the early universe, releasing far IR radiation from near the photon sphere (note, this isn't Hawking radiation, just delayed and redshifted background) that an ultraboosted observer should be able to spot.
(Work in this area is usually about considering what an ultraboosted massive particle does as it passes a massive observer. For example, in the 1970s the Aichelburg-Sexl ultraboost tried to capture what a neutral cosmic ray would do -- gravitationally -- to a spherical gas cloud along its path. In essence there is an exchange of momentum: the ultraboosted particle loses some, and small parts of the gas cloud gains some. Increasing the mass and structural complexity of the ultraboosted observer inevitably runs into inelasticities. That admits a hand-wavy expectation that even below the threshold at which subatomic physics of the ship becomes relevant, the ultraboosted spaceship is liable to become so very hot that its thermal spectrum masks practically all the light from everything ahead of it.)
As you raised in your final question, there is good reason to believe there is plenty of very deep IR electromagnetic radiation out there from all sorts of sources. Most models of cosmic inflation will produce gravitational and electromagnetic radiation with all manner of wavelengths, many of which are stretched to cosmological lengths. We don't really have a good view of any of that because presently we can't yet make a sufficiently cold ultra-long-wavelength (sub-30 MHz; cf. FARSIDE, LCRT, GO-LoW) detector and don't even have good ideas about how to look at astrophysical sources below kHz. There's probably lots of bodies (e.g. magnetic interactions with exoplanet atmospheres and gas clouds) that will become optically bright as the CMBR is blueshifted out of the range of human vision.
The visualization also does not offer up a physical observer who would feel variations in the galactic magnetic field (cf. the Sokolov-Ternov effect with enough "nines"), the cosmic neutrino background, and two-photon physics from ultraboosted CMBR and starlight interacting with the observer's photon shock front.
The visualization is pretty, it captures some gross aspects of being relativistically boosted, but as many scattered threads in the discussion discuss, it's probably not a useful hint about what one of our descendants might experience if somehow they could reach "a few nines".
Yes, that's the microwave background being blueshifted, but why is it growing across the screen? Shouldn't the angle that's blueshifted be less and less as you go faster? At lower speeds we do see the starbow effect, passing stars going down the spectrum as the angle increases.
And what becomes of the stars as you're going fast enough? Yes, most of the energy is shifted out of view but don't stars emit some at such low energies?