I suppose there could be all sorts of reasons for this:
Working class people do more manual jobs, hence accidents at work, or long term injuries from manual labour leading to early death (miners for example).
Working class people may have poorer diets, either through buying poor quality food or fast food, also high fat food and so on. Lack of education about food amongst working class may also play a part.
Working class people MAY be more likely to smoke and drink, leading to early death ( I know affluent people can also drink and smoke to excess).
Poor housing also probably contributes to early death (colder houses, sanitation etc).
Working class people are less likely to be able to afford extra health care, or to be in a private medical scheme like BUBA.
While a generalization I would guess affluent people are overall better educated so may be quicker at getting medical advice when they are ill, or looking after their bodies as they get older.
I guess this statistic is probably true in every country in the world, and no amount of money or education is ever going to change it.