But they do not cling to their body, so why eat? They don't cling to existence, and they don't cling to helping (to whatever extent that is possible) others either. Without clinging to the body (in order to...), one would die and be unable to help others.
As I understand it, certain perception of self is required to respond to stimuli such as hunger, heat, cold, danger, etc. One protects oneself and one responds accordingly because one knows that "I am in danger or discomfort therefor I, not someone else, need to do something". IMHO
But why can't arahants eat after their enlightenment? Starving oneself to death on purpose is the same as killing a sentient being, thus breaking the 1st Precept, something an arahant wouldn't do..