The brain change that they are talking about is most likely the down-regulation of dopamine receptors in the brain’s reward center—the same change that happens with any addictive drug taken in quantity. Dopamine stimulates the cells, and too much dopamine can damage or even kill them, so each hit of dopamine causes these cells to down-regulate their dopamine receptors in self-defense.
I don’t know if ketones in the brain will cause dead cells to be replaced or help damaged ones to heal, but it is possible that the absence of high levels of dopamine is enough to allow healing and new cell growth.
The help with depression is the result of higher levels of seratonin in the brain, they think. Serotonin is made of tryptophan, one of the essential amino acids. The problem is that serotonin is useful in many places in the body, so only about 10% or so of the tryptophan we eat even makes it into the brain.