Why was the Angel Michael fighting the Devil for Moses' body?

This is generally thought to refer to:

Zechariah 3:1and 2 Then he showed me Joshua the high priest standing before the Angel of the LORD, and Satan standing at his right hand to oppose him. 2 And the LORD said to Satan, "The LORD rebuke you, Satan! The LORD who has chosen Jerusalem rebuke you! Is this not a brand plucked from the fire?"

Where the body of Moses enters the picture is:

Deuteronomy 34:5 through 39 So Moses the servant of the LORD died there in the land of Moab, according to the word of the LORD. 6 And He buried him in a valley in the land of Moab, opposite Beth Peor; but no one knows his grave to this day. 7 Moses was one hundred and twenty years old when he died. His eyes were not dim nor his natural vigor diminished. 8 And the children of Israel wept for Moses in the plains of Moab thirty days. So the days of weeping and mourning for Moses ended. 9 Now Joshua the son of Nun was full of the spirit of wisdom, for Moses had laid his hands on him; so the children of Israel heeded him, and did as the LORD had commanded Moses.

Moses is the acknowledged author of the Pentateuch, but he obviously was not the one who described his death. Deuteronomy 34:5 through 39 is attributed to Joshua by most Bible students. And if we go back to chapter 3 of Zechariah we find that it is about Joshua and God anointing him to carry on for Moses. I will not quote it here for the sake of brevity, but you can read it for yourself.

Hope this helps.

In addition to Bye's good answer, I would like to add the following.

Uptill this point only one other person has ever ascended to heaven (from the written record) - that was Enoch. The major difference between Enoch and Moses was that Moses was resurrected from death while Enoch was translated alive.

Paul says in Romans,

Romans 5:14, 17: Nevertheless death reigned from Adam to Moses, even over them that had not sinned after the similitude of Adam's transgression, who is the figure of him that was to come... For if by one man's offence death reigned by one; much more they which receive abundance of grace and of the gift of righteousness shall reign in life by one, Jesus Christ.)

Death reigned from Adam to Moses - The power of death over the human race was hitherto unbroken. Death seemed to rule with an Iron Scepter. Satan who is the father of sin and thus the originator of death apparently had that unbroken power - Once you die, no matter how true you were to God, you belong to Satan (as everyone who sins are children of satan).

But now - Wow, death's power was broken! and a captive of Satan was being led away by Michael toward heaven. If this would happen to everyone who would obey God.... Satan could not permit it.

That's why he contends when Michael came to retrieve Moses.

Michael fought over Moses' body because Moses needed his body still to appear to Jesus in the future, when he appeared on Mount Hermon (Matthew 17:1–4). Enoch had his body too: he was taken alive with his fleshly body.

The spirit of Jesus did not appear to the Apostles alone: Jesus had his body too.(Luke 24:37–43)

My reading of the Navarre New Testament Compact Edition's note to Jude 8-13 (RSVCE) is that the devil wanted to use Moses' body to incite the children of Israel to sin [idolatry].

To Illustrate the sinfulness of these offences against the angels, the sacred writer uses a popular legend recorded in the apocryphal Assumption of Moses, according to which, when St. Michael was preparing to bury Moses' body, the devil tried to wrest it from him. St. Michael prevented him from doing so, but he did the devil no hurt; he simply appealed to the judgment of God.

The writer provides three further biblical examples to show the evildoing of the false teachers - Cain (Gen 4:3), Balaam (Num 31:16; Rev 2:14; cf. 2 Pet 2:15), and Korah and his followers who rebelled against Moses (Num 16).

The false teachers are quite happy to attend Christian assembles, but they lead an immoral life and cause scandal.

This is confrimed by

St. Michael concealed the tomb of Moses; Satan, however, by disclosing it, tried to seduce the Jewish people to the sin of hero-worship. St. Michael also guards the body of Eve, according to the "Revelation of Moses" ("Apocryphal Gospels", etc., ed. A. Walker, Edinburgh, p. 647). - Source: St. Michael the Archangel | New Advent.

It appears that the author of Jude was quoting from an apocryphal Jewish scroll known as the Assumption of Moses.

Origen stated in De principiis, III,2,1:

"We have now to notice, agreeably to the statements of Scripture, how the opposing powers, or the devil himself, contends with the human race, inciting and instigating men to sin. And in the first place, in the book of Genesis, the serpent is described as having seduced Eve; regarding whom, in the work entitled The Ascension of Moses (a little treatise, of which the Apostle Jude makes mention in his Epistle), the archangel Michael, when disputing with the devil regarding the body of Moses, says ..."

A fragment of the Assumption of Moses has been discovered, but it does not include the account of Michael and the devil disputing over the body of Moses, which means that we can not say how this account played out, or why the author of Jude chose to mention it, apart from its apparent support for Jude's theology.