Why is quixotic pronounced as it is?

/'kwɪksət/ is clearly an anglicization of the Spanish spelling.
It's equally clear that such spelling pronunciations have always been very common.

This one, in particular, can be seen in action at the very end of
Canto 13, Stanza 10 of Byron's Don Juan
  (a title, incidentally, pronounced /dan'dʒuwən/ by the author, as the poem makes clear)

Redressing injury, revenging wrong,
  To aid the damsel and destroy the caitiff;
Opposing singly the united strong,
  From foreign yoke to free the helpless native: --
Alas! must noblest views, like an old song,
  Be for mere fancy's sport a theme creative,
A jest, a riddle, Fame through thick and thin sought!
And Socrates himself but Wisdom's Quixote?

Note the rhyme: wrong - strong - song alternating with caitiff - native - creative,
and ending in a couplet, with its rhyme boldfaced above. In this couplet,
Quixote has to be pronounced in two syllables, not three,
and the last two syllables rhyme with thin sought.

I.e, /'kwɪksɔt/. And from there to /'kwɪksət/ is no distance at all in modern English.

It appears to be dialect issue. Its not a word that comes up much in daily conversation, but the few people I know who use it pronounce it closer to your second (your supposedly "correct") way. More like KEY-hot-ick (emphasis on the first syllable, short o in the middle). However, there are comments below to the effect that your first pronounciation is reported as the "correct" one in the OED, and is understood as such in the UK.

The people I have heard use it are American Midlands dialect speakers (both Northern and Southern varieties). I suppose its possible that your dialect area (from your user info, I'm guessing California English?) tends to use the more normal Anglicized version.

Probably the reason for that first reported usage is that it is the pronunciation an English speaker would naively expect for that assemblage of letters, in the absence of any other information. The most common word starting with "qui" is "quick", which is pronounced with the same "kwi", and an "X" in English usually produces a "ks" sound. So if you didn't know the word derived from the name of a character in a Spanish novel, you'd expect it to be pronounced your first way, not in a Spanish-influenced way.

I've often wondered about the pronunciation of this word. Where I live in New England, we pronounce any Spanish word that has an X, such as "Mexico," in the anglicized version, unless we are speaking in Spanish. For that matter, when we refer to Cologne, Germany, we don't say "Köln, Deutschland," either, unless we are speaking in German. So why would we consider saying "Kee-hot-ic?"

My hunch is this: We all know the Spanish pronunciation of "Don Quixote" as "Don key-ho-tee" from having heard the name pronounced in story-telling, or in the musical, "Man of la Mancha." So perhaps some of us think we ought to be consistent and also use the Spanish pronunciation for the adjective. But that makes a strange amalgam, because Spanish adjectives don't end in "ic." The actual Spanish adjective is "quijotesco."

So, if we are going to use the English adjective form, then I think we should be consistent and use the English pronunciation.

Other answers have covered the bases on quixotic quite well, I think. I just want to note here that quixotic isn't the only Spanish-derived word containing the letter string qui that (at least some) British English speakers pronounce in a very un-Spanish way.

In the United States, children are taught in elementary school about the Spanish conquistadors, who explored (and of course conquered) large areas of the New World during the very late 1400s and the 1500s—and the only pronunciation I have ever heard used here in the United States for the word conquistador is kunkeestuhdor, with the accent on the kees syllable.

But as this video of Procol Harum performing its song "Conquistador" in 1977 demonstrates, Gary Brooker, the band's (English) lead singer, pronounces the word kunkwisstuhdor, with the accent in the kwiss syllable. This, it seems to me, is entirely consistent with pronouncing quixotic as kwiksottik with the accent on the sott syllable.

Indeed, if we accept the three letters qui in Spanish as consistently receiving the pronunciation kwi (with a short i sound) in certain forms of British English, it seems to me that pronouncing the following x in quixotic as ks rather than as h follows as a matter of course, given the strangeness (to the mouth of an English speaker) of pronouncing the h sound immediately after a short i sound. In English we have several words that involve pronouncing an h immediately after a long e sound (beehive, knee-high, freehold, and hee-haw, for example), but none I can think of that involve a short i immediately followed by an h sound.

For that reason, I would expect an English speaker attempting to pronounce frijoles either to try to match the Spanish pronunciation with something like freeholeez (with the accent on the ho syllable) or to keep the i short and pronounce the word something like frijoleez (with the accent on the jo syllable) or frijolz (with the accent on jolz).