It always strikes me as weird that people would be shocked that a substance commonly prescribed to affect the brain... affects the brain.
Beyond increasing BDNF activity (which promotes neuron survival and new growth), one of the main therapeutic targets is Glycogen synthase kinase-3 beta, which controls a wide range of developmental parameters for neuronal development, including discouraging regeneration and promoting apoptosis, and is pro-inflammatory (CNS inflammation is itself associated with Alzheimers). Lithium reduces its activity, both with direct and indirect inhibition. While GSK-3B is essential to a degree, overactivity of GSK-3B is associated with a wide range of neurodegenerative diseases, including Alzheimer's.
Lithium also helps promote cleanup (misfolded proteins, degradation products, etc) via autophagy by reducing the activity of IMPase... at least at low doses. At high doses, it can cause the inverse effect, due to its GSK-3B and mTOR impacts. In general, though, it seems to typically be pro-autophagy.