Why was the dead body of Saul cremated?

Why was the dead body of Saul cremated?

Simply put it was a time of war and they probably feared that the Philistines might try to desecrate the bodies of the fall men of Israel. Cremation was not a Hebrew practice. Another possibility would be that the bodies started to smell due to putridication or infection.

Cremation was not a Hebrew practice, but in the present case was probably adopted to avoid the possibility of further insult to the bodies, if the Philistines should take Jabesh. The condensed account in Chronicles omits the mention of the burning. - Cambridge Bible for Schools and Colleges

This “burning the corpse” was never the custom in Israel, and was restricted to criminals convicted of a crime of the deepest dye (Leviticus 20:14). The Jews in all cases buried their dead. The Chaldee therefore interpret the words relating this act of the men of Jabesh-Gilead, in the case of Saul and the princes, as referring to the solemn burning of spices, a ceremony which was afterwards performed at the burial of some of the kings of Judah. (See 2Chronicles 16:14; 2Chronicles 21:19; Jeremiah 34:5.) But the language used in these cases is different; here it is expressly stated that “they burnt them.” The reason for their thus acting is clear. The mutilated trunks had been exposed for some days to the air, and the flesh was no doubt in a state of putrefaction. The flesh here only was burned. The bones (see next verse) were reverently and lovingly preserved, and laid to rest beneath the friendly shade of the great tamarisk tree of Jabesh. - Ellicott's Commentary for English Readers

Valiant men arose, and went all night, and took the body of Saul and the bodies of his sons—Considering that Beth-shan is an hour and a half's distance, and by a narrow upland passage, to the west of the Jordan (the whole being a journey from Jabesh-gilead of about ten miles), they must have made all haste to travel thither to carry off the headless bodies and return to their own side of the Jordan in the course of a single night.

burnt them - This was not a Hebrew custom. It was probably resorted to on this occasion to prevent all risk of the Beth-shanites coming to disinter the royal remains for further insult. - Jamieson-Fausset-Brown Bible Commentary

All the valiant men arose,.... Of the city of Jabeshgilead, fired with indignation at the Philistines' ill usage of Saul and the bodies of his sons:

and went all night; not only for secrecy, but for haste:

and took the body of Saul and the bodies of his sons from the wall of Bethshan, and came to Jabesh; brought them with them thither, the Philistines either having no knowledge of it, or not daring to oppose them:

and burnt them there: that is, the flesh of them, for the bones they buried, as in 1 Samuel 31:13; and this they did, contrary to the common usage of the country, which was not to burn; but this they did, that if the Philistines should come to recover them, they would not be able to do it: though the Targum is,

"they burnt over them, as they burn over their kings there;''they made a burning for them of spices over them; or of their beds, and other household goods, as Kimchi and Ben Melech observe, though they prefer the other sense; see 2 Chronicles 16:14 perhaps the true reason might be, because they were putrid and infectious. - Gill's Exposition of the Entire Bible

Probably because his body had already started rotting. It is possible that there may have been a few days between his death and the recovery of his body by the men of Jabesh-Gilead. I can't say that for sure but it is a possibility.

1 Samuel 31:8-13
⁸ It came about on the next day when the Philistines came to strip the slain, that they found Saul and his three sons fallen on Mount Gilboa. ⁹ They cut off his head and stripped off his weapons, and sent them throughout the land of the Philistines, to carry the good news to the house of their idols and to the people. ¹⁰ They put his weapons in the temple of Ashtaroth, and they fastened his body to the wall of Beth-shan. ¹¹ Now when the inhabitants of Jabesh-gilead heard what the Philistines had done to Saul, ¹² all the valiant men rose and walked all night, and took the body of Saul and the bodies of his sons from the wall of Beth-shan, and they came to Jabesh and burned them there. ¹³ They took their bones and buried them under the tamarisk tree at Jabesh, and fasted seven days.