Basically we experience a constant acceleration in our frame of reference relative to the Earth. Some energy source is accelerating the Earth (although we are magically shielded from this energy). The Earth is moving away from this source at near the speed of light, and the source does not observe any significant acceleration due to the force it exerts, but we relative to the Earth still feel an acceleration.
The only reason we would experience any acceleration at all is if the Earth is being accelerated by this source (the UA) and not us. To me this is the main problem with the UA, because it is really not "universal". It accelerates the Earth and the visible stars, but not anything else on the Earth that we can observe! Even when airplanes fly high into the atmosphere they experience gravitation the same and thus must still be shielded form this UA.
We can speculate that some semi-infinite energy source could be accelerating the Earth relative to some reference frame that we are moving away from at near the speed of light, this is in a way similar to the proposition of Dark Energy. The real kicker is the requirement that this dark energy be selective and ignore pretty much every other piece of relevant local matter around the Earth that we can observe, which kills its credibility.