How does the wife specifically die in The Road?
After the unspecified natural disaster destroys society as they know it, the world of the characters in The Road spirals downward until cannibalism and crime is the order of the day. Our main character, his wife and child have survived up until now, but the wife has grown despondent. She knows eventually she will be captured, assaulted, and then eaten, and she cannot live with the fear.
Her husband has tried to prepare her for this by teaching her to slit her wrists with obsidian, even teaching her how to do it most effectively so that she can avoid her most feared fate if it comes to that. Without knowing it, he has taught her how to commit suicide before they are ever captured. He tries to talk her out of it, but she is adamant, and the man and their son walk away knowing she cannot be convinced not to kill herself. While the book doesn't detail her death beyond that, it's clear that's what happens.