He killed a lot of his brain cells and became terminally stupid. The usual way to say this is "The alcohol fried his brain"

Most likely (1). There are many people who are "brain dead", however their bodies continue to function normally. Hence you may "burn out your brain", without killing your body. The above sentence, however, is most likely an exaggeration, meant as a way of insulting a person; implying that they have become stupid after consuming too much alcohol.