I get why murder and idol worship are things you have to give your life up for. But why is giluy arayos - sexual sin so fundamental as to require a person to give up one's life? I'm looking for a taamei hamitzvot perspective, not just the formal derivation of the law.
He says that the point of these laws is to tell us that the central symbol of our service of God, the Altar, has to represent the fundamentals of our actual service - our adherence to the three fundamental laws. We therefore have to, through the laws given here, "expel the very last trace" of the fundamental sins from the Altar, just as we do so from our actual lives.
He formulates the Big Three as "the culmination of sin against God, of sin against one's fellow man, and of sin against oneself."
So, how does sexual immorality constitute "the culmination of sin against oneself"?
I think R' Hirsch touches on this in many places, but I found it set out pretty clearly in his commentary to Leviticus 18:6, at the beginning of a list of couplings particularly prohibited to Jews:
(my bracketed comments where indicated)
To our purpose, I think what he's saying is that sexual morality is the paradigm of human adherence to God's Law through choice. A person who violates this Law turns his back on his mission as a human being, just as an apple tree that somehow decided to produce grapes instead would be turning its back on its mission as an apple tree.
Just to note, 'Giluey Arayos' includes two different forms of sin: incest and adultery. They may have different reasons.
Most societies (agreed, not all) have held these two acts to be contemptuous or taboo, which may mean that it's an area of morality which should be self-evident, and if it isn't obvious, then you should check your moral intuition. I personally am not inclined to such a view (though I may be guilty of tainting myself by taking college courses in ethics). However, I believe that R. Yitzchak Arama (Vayikra Shaar 64), R. Moshe Feinstein (see Y.D. 3:115 where he discusses homosexuality) do think so, as well as other great Rabbis.
Ramban (Vayikra 18:6 and 18:29) alludes to a 'hidden' reason, but says that ultimately, these prohibitions are gezairos melekh, but unintelligible to humans. His 'hidden' reason is probably something along the lines of Rabbeinu Bachya's mystical explanation (ibid.) because the Ramban seems to use similar terminology regarding kilayim.
Rambam in Moreh Nevuchim 3:49 states that the reason for sexual prohibitions are to limit indulgence in said act. While this may not be so satisfying an answer as to why it should be punished so severely, bear in mind that the Rambam believed in asceticism as a prerequisite to intellectual growth, which is the point of the entire Torah. The main point of asceticism is that a person should not even think about these matters at all, to keep his mind focused on lofty things. Therefore, he writes, the Torah gave arayos penalty of death to stop a person from even thinking of committing these sins. Ibn Ezra to Vayikra 18:6 seems to allude to this as the reason as well.
The Rambam in the same passage there states that the prohibitions of incest and adultery also prevent one from acting brazenly towards the women whom he should respect. Sefer HaChinuch (Mitzvah 35) believes that this is a separate reason and likes it better than the first. See below for further elaboration.
Rav Saadia Gaon (Emunos Vedeios 3:2) begins with a similar reason as the Rambam, but adds that incest is prohibited because if men were allowed to marry their sisters, then men would marry only their most beautiful sisters while the other girls would be shunned by all other men because even their own brothers didn't like them enough to marry them. His explanation may sound a bit odd and far-fetched to the modern ear though, but reading this gave me an idea.
If I may elaborate on and modernize these Medieval approaches, and combine it with the sense that I get from Maharal (Nesiv Hateshuvah 2) as well as the commentary of R. S. R. Hirsch (Vayikra 18:6), I think the picture that emerges is as follows. The most important social institutions for all aspects of moral and religious development is the family. Violations of both adultery and incest violate this familial structure. Marriage is a sacred partnership, where two people help each other fulfill God's role for humankind, and one that necessitates exclusivity (polygamy notwithstanding). On a different level, Rambam and R. Saadia Gaon point to the fact that family members need to act a certain way towards each other, and if they don't, if they taint their family structure, they lose their ability to relate to all of humanity with the proper dignity. These structures that are at the core of a person's relationships create the framework for treating both man and God as necessary.
there is a chazal that arayot and kedusha (holiness) cannot co-exist.
since the purpose of torah is to induce a person to sanctify himself, then transgressing arayot is becoming the diametrical opposite of what God wants.
BTW, this is why the gedolim are so scared about the internet and have labeled it the greatest problem of all time - due to all the pritzut one can easily access, and this ruins a person from the inside.
one who gets involved in such things can become irreparably corrupt as we saw by the flood whose main sin was arayot.
After a careful reading of the Talmudic explanation for this law (in Bavli Sanhedrin 74), it is clear that the "giluy arayos" that is included in the "big 3" is, specifically, rape. This is the context in Devarim 22:26, which says that rape is as bad as murder. When a man rapes a girl, he actually murders her, therefore it is a mitzva to save the girl by killing the raper.
Therefore, if someone tells you "either rape that girl, or I kill you", then it is just like saying "either kill that girl, or I kill you" - in both cases you should be killed and not rape/kill.
(I am aware that in the Yerushalmi the explanation is different... this is a topic for another discussion).