This article suggests that inflammation in response to the surgery is the problem, but as far as I know, anesthesia itself is very dangerous. It can reduce the amount of oxygen the brain is getting for several hours, and if the brain doesn't get enough oxygen, neurons can die. Some of the cognitive changes following anesthesia could be brain tissue death due to the anesthesia itself. Careful dosage and monitoring is required, and you should probably get local anesthesia if possible.
Anesthesia kills brain cells, but what does it mean? https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5119529