Questions of why are often difficult to answer when it comes to language. Over time, people come to express things a certain way, and what begins as a personal preference or novelty may over time become convention. Anyone with small children has probably been asked why we say inside-out instead of outside-in, and responded That's just how we say it, sweetheart. Now stop changing the subject and finish your asparagus.
That said, there is a large vocabulary of terms for the multitude of shortening phenomena in languages, from clipping to contraction, ellipsis to elision; human communication is full of aphetisms and hypocorisms and abbreviations and univerbations. It seems the inclination to reduce speech to the absolute minimum needed to express an idea unambiguously is quite human and quite universal, else would we need so many words for it ;) ?
When it comes to numbers, we use explicit values where they are important, as in science and industry. But most of the time, we will understand the order of magnitude of the numbers being discussed, and can often drop them. This is stereotypically associated with Americans, and among Americans with obnoxious young workers in finance, but if a salesman tells me I can give it to you for fifteen, I understand that the car salesman means $15,000, the refrigerator salesman means $1500, and the grocery checkout clerk means $15. Since most of us live under 100 years, we'll similarly understand what century is meant. Someone wearing a Class of '21 sweatshirt will be understood to be either 2021 or 1921, with no chance of confusion for the other. The '68 Ford Mustang does not refer to a horse ridden by a Mr. Ford in 1468.
And while the Gregorian year is supposed to correspond to years since the birth of Jesus of Nazareth (as [mis]calculated in late antiquity), in practice, even among devout Christians, it is a label, not an anniversary counter. There is no theological profundity associated with the year counter (at least since the first millennium). So although legal documents and other formal communications will spell out the full year (in the Year of Our Lord One thousand Nine Hundred and Sixty-five and in the thirteenth Year of Our Reign), there is no quarter of society which demands its use elsewhere.
So if linguists tell us that humans have a tendency to shorten words and sentences, and if we acknowledge that the long form is neither necessary for understanding nor required by fixed phrasing, and we observe that nineteen ninety-three has half the syllables of one thousand nine hundred [and] ninety-three (and the dropped syllables are in the almost-superfluous century indicator), there's really no question here. Mitch gave you the answer, using a few thousand fewer keystrokes than I have: it's shorter.
