Well, it just ain't true. According to a University of Washington study, healthy foods do not cost more than junk food. Sure, there are some very expensive healthy foods, but most of the basic staples (rice, beans, chicken, carrots, etc.) are much less expensive than fast food and are plenty nutritious.
So why do the people eat so badly if it's not because of cost? According to the article, it's time and convenience. It takes 9-16 hours a week to shop for groceries and cook and serve them at home. In households where all the adults work, nobody wants to take that kind of time, so they eat out, and fast food is cheaper than other kinds of restaurants.
In other words, these people are not eating fast food because they aren't making a lot of money. If they suddenly make more money, they will still not want to go grocery shopping or cook meals for the family. I suppose some might start going to more upscale restaurants, but most families are going to require a really huge raise if they want to eat like that every night.