Like: "quite the singer", "quite the writer", etc. while he/she is just a singer/writer and is not the only singer/writer, etc in that context.
Quite a (and quite the, sometimes used ironically) indicates that the specified thing or thing is recognized as notable, remarkable, or impressive.
Quite a party, isn't it?
It has been quite a year.
Quite the little horsewoman, aren't you?
Quite the thing is a dated way to say socially accepted.
She was quite the thing in heels and stockings and lipstick.
[The examples are taken from the NOAD]
As noted by @ColinFine, "quite the X" is simply an idiom. It's not any sort of standard grammatical construction and so cannot be explained according to standard grammatical rules. Other idioms that include inexplicable instances of "the" include "what the hell," "the hell you say," and to be "for the asking."
If 'life is context', then think about why this phrase has become so over-used during the past 2-3 years. To me, the increased use of 'quite the' rather than 'quite a' is to use the phrase to appear smarter or more 'hip'. In America, we are continually hearing commercials with phrases meant to imply that a 'hip' or 'in the know' person talks a certain way. And the commercial message is to get the listener to do what is wanted by the vendor.