means that no one reacted to something, as in I told my friend to never call me again, but he didn't bat an eye my friend did not show any reaction to what I said to him

It literally means to wink or to blink. Sometimes, when a person is trying not to show a any visible reaction, they won't change their expression, but can't avoid blinking. That is why "didn't bat an eye" means "showed no reaction at all."