Here both what it means and how it comes will be elaborated as follows.
Actually, the meaning of 啃老族 is the same as the abbreviation NEET to some extent. So I want to illustrate 啃老族 with the aid of NEET. This abbreviation refers to the people who has already graduated, reached the working age and able to work, but still Not in Employment, Education or Training. (For more details about this abbreviation, just check it out in Wikipedia
http://en.wikipedia.org/wiki/NEET)
Now let's try to get the meaning of 啃老 first before knowing 啃老族.
Those people NEETs has no work and therefore has no money. To survive, they have to get money from their parents. So 啃老 refers to the action of getting money.
But what is the relation between the 啃老 and this action?
The word 啃 means eat, and the word 老 refers to the older. 啃老 just like eating the flesh of the older. So actually it's a metaphor to describe the action of getting money from parent.
So what does 啃老族 mean?
It's easy to answer. It simply refers to the group of NEETs or say the group of people who 啃老.
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Hope these illustration make sense. In case some confusing points still remain, just feel free to let me know.