Dr. Jay Goldstein's theory is that our whole problem comes from information gating problems in the brain. As he says in Betrayal by the Brain, "If input processing is not properly controlled, output will not be correct." If information gating doesn't work properly, the signal to noise ratio is lowered, and the ability to determine salience is impaired. This applies to both the cognitive centers and the parts of the brain that regulate the body.
As the GABA receptors are inhibitory, it's pretty straightforward to see how modulating them can help with sleep. As for clarity, one of the things that the GABA receptors inhibit is the NMDA receptors. If these are hypersensitive, as they are in people with ME/CFS, then their output will be very noisy. Inhibition of the NMDA receptors will increase the signal to noise ratio in such a circumstance, and this is experienced as greater clarity.
Dr. Cheney also noticed this effect, although he did not provide an explanation. But he did mention that taking a small amount of Klonopin during the day can actually increase energy. This is just a slightly different effect of improving the signal to noise ratio. With a stronger signal and less noise at the NMDA receptors, the brain works better, and is more able to control the body properly. Essentially, everything works more as it should, and this includes proper maintenance of energy levels, which is simply not possible with a low signal to noise ratio.