I assume they are talking about moving from "working class" to "middle class" (N.B. It's not really possible to move from "middle class" to "upper class" unless

I assume they are talking about moving from "working class" to "middle class" (N.B. It's not really possible to move from "middle class" to "upper class" unless by marriage and then it's only really your kids that would move class bracket. In England, accent/grammer usage is a strong indicator of class (as I assume it may be in other countries).