Why do we use the definite article in the expression "quite the [noun]"?

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]}

Stating "Quite a [noun]" indicates that one is notable, while stating "quite the [noun]" is remarkable.

It is mostly the difference between the indeterminate (a—or, just another [noun]) and the determinate (the—a specific [noun]).

We use the definite article to show uniqueness.

This case, saying "quite the singer" suggests that the singer is unique in some way -- e.g. being noticeably more talented than the others.