The first thing to understand is the intent of the passage.
In I Samuel 19:20 Saul sends a posse to arrest David. This posse meets Samuel and his band of prophets and the spirit of God rests on the posse and they become prophets themselves. Saul sends a second, and then a third posse with the same result. Finally Saul himself goes to Nayot and he also becomes a prophet under Samuel.
This story emphasizes the view that God himself is on the side of Samuel and David and is thwarting Saul's intent. Saul prophesizes, but involuntarily in humiliating and comic circumstances.
The narrative continues, "Wherefore they say, Is Saul also among the prophets?". The Hebrew for this is:
"על כן יאמרו, "הגם שאול בנביאים
The difference between this saying phrased as a question and the same saying phrased as an assertion "Wherefore they say, 'Saul is also among the prophets'" is one letter only, the ה in the word הגם. It is likely that the author added this single letter to mock an original Benjaminite, pro-Saul faction saying that was phrased as an assertive - without the letter ה.
Placing this bent saying in the context of the above comic narrative gives it a further mocking connotation: yes, maybe Saul was a prophet, but only to get him temporarily out of the way.
After understanding the intent, we can look into the detail of the clothes, which adds further insult to the image of Saul.
The cultural context is 2 Samuel 6:20, Then David returned to bless his household. And Michal the daughter of Saul came out to meet David, and said, “How glorious was the king of Israel today, uncovering himself today in the eyes of the maids of his servants, as one of the base fellows shamelessly uncovers himself!”
This is Saul's daughter expressing the values she learned from her father, that for a king to disrobe, even partially, for any reason, is a humiliation. Michal is in effect expressing her father's feeling about what happened to him I Samuel 19:23.
The meaning of ערם "nakedness" in the OT is closer to "uncovered" than "naked".
Isaiah 20:2 NKJV: at the same time the Lord spoke by Isaiah the son of Amoz, saying, “Go, and remove the sackcloth from your body, and take your sandals off your feet.” And he did so, walking naked and barefoot.
2 Kings 1:8 NKJV: So they answered him, “A hairy man wearing a leather belt around his waist.” And he said, “It is Elijah the Tishbite.”
Prophets were then often poor people living on the fringes of society, without the means to acquire respectable clothing and so the public perceived them.
See also Zechariah 13:4: And on that day every prophet who prophesies will be ashamed of his vision, and he will not put on a hairy cloak in order to deceive.
and Matthew 3:4: Now John himself was clothed in camel’s hair, with a leather belt around his waist; and his food was locusts and wild honey.
and Genesis 39:12: that she caught him by his garment, saying, “Lie with me.” But he left his garment in her hand, and fled and ran outside.