Parable of the Prodigal Son
I think that the way to understand this idea (that the prodigal son was lost and dead) is think that he was lost and dead in terms of his family and his morals.
When the son was gone, he was dead to his family because he was not with them. More importantly, he was dead and lost to his morals -- he had forgotten how to behave in good ways. When he came to his senses, it was like being born again. He realized that he had given up a good life for a bad life and he understood that he should go and beg forgiveness. He was willing to try to start a new life (after having been dead) by being a hired hand instead of an honored son.
In the parable of the Prodigal Son, the older brother becomes angered when he hears music and dancing when he comes from the field. Seeing that a feast with a fattened calf has been prepared for this wayward brother and that he has been given a new robe, the older son tells his father that he has worked all the years that the sibling has been gone. Deeply hurt, he tells his father that he has "been slaving and never disobeyed" any of the father's orders, yet he was never even been given a goat for a celebratory feast.
The father explains that the older son has always been with him as part of the family. But, the wayward son has been lost since he left and did not contact the father while he was gone. For, he only thinks of his father's place as he considers where he can find work after the famine where he lives. In the father's words he has been "dead," and when the father sights the younger son coming a long ways off on the road, the father "is filled with compassion," and he forgives his prodigal son whom he has not thought he would ever see. After having taken his inheritance, the younger brother severed all ties with his family. In this manner, the prodigal son has been "dead" and has now "been found" as the father discovers him as he walks up the road. Elated that he has returned, the father calls for a feast.