Shared Tree History

Rashi: "For a hanging [human corpse] is a blasphemy of G‑d"
This is a degradation of the [Divine] King in Whose image Man is created, and the Israelites are G‑d’s children. This is comparable to two identical twin brothers. One [of them] became king, while the other was arrested for robbery and hanged. Whoever saw him [the second brother, suspended on the gallows], would say, "The king is hanging!"

Baal HaTurim: "on the gallows"
...the gallows on which the corpse is hanged is buried with him. Right next to this verse is: "rather you shall surely bury him", to indicate that the gallows on which the corpse is hanged is buried with him. It also implies that he is buried in a coffin.

Targum Yonatan: When a man has become guilty of the judgment of death, and is condemned to be stoned, and they afterwards hang him on a beam, his dead body shall not remain upon the beam, but he shall be certainly buried on the same day; for it is execrable before G‑d to hang a man, but that his guilt gave occasion for it; and because he was made in the image of G‑d, you shall bury him at the going down of the sun, lest wild beasts abuse him, and lest you overspread your land, which the L-rd your G‑d gives you to possess, with the dead bodies of criminals.

Mishmeret Itamar: This suggests the effort towards compassion for the perpetrator of the crime. Even severe crimes come down through the generations as a result of the fracture caused around the Tree of Knowledge. As fracture generates crack which in turn generates split and things follow in train, ultimately this person came under the effect of the tree and caused whatever he caused. Hanging the person on a tree is to remember that this crime was not entirely their own fault but also an inheritance of the fractured consciousness, the tree history we all share. (Peninei HaChassidut)

Ohr HaChayim: This is a commandment address to the people at large not to suspect a Torah scholar of a sin when they see him do a deed that could be interpreted as a sin. You must assume that even if he had already done a sin, he has repented of it. "His dead body" is a reference to the sin. You must bury this sin on that very day when it had been committed or presumed to have been committed. If you were not to give the scholar in question the benefit of the doubt and at least assume he has meanwhile repented his mistake, you would be desecrating the Name of G‑d and would besmirch His honor.

Ramban: "do not defile your land"
This is not merely a reason. According to the Rabbis it is a second negative commandment. It is because of this commandment that Joshua buried the Canaanite kings during the day of their execution.

Zohar Emor 88:
After the soul has left the body, and it remains spiritless, it is forbidden to leave it unburied, as written, "his body shall not remain all night upon the tree, but you shall surely bury him that day." For a corpse remaining unburied for 24 hours, a day and a night, causes the parts of the Chariot to weaken, and detains the actions of the Holy One, blessed be He, from being carried out. For the Holy One, blessed be He, may have decreed upon him another incarnation at once, on the very day he died, in order to help him. But as long as the body is not buried, the soul does not come before the Holy One, blessed be He, nor can it be in another body in another incarnation, for a soul is not given another body until the first one is buried. This resembles a man whose wife died. He is not qualified to marry another wife before he buries the first one. Hence the Torah said, "his body shall not remain all night upon the tree."

Another explanation: when the soul has left the body and wishes to go to that world, it may not enter it until it is given another body of light. Then it can enter. You may derive this from Elijah, who had two bodies, one in which he was seen by people below, and another in which he was seen above among the celestial holy angels. As long as the body is not buried, the soul suffers and the spirit of defilement is there to dwell upon that body and defile it.

Since the spirit of defilement is in readiness one must not keep the body for a night, because the spirit of defilement is present at night and spreads, throughout the land to find a soulless body to defile it. It is defiled even more. Therefore it warned the priests, saying, "There shall none be defiled for the dead among his people". (Levit. 21:1) Since they are holy, the spirit of defilement will not dwell upon them and they shall not be defiled.