Theology student here!

First I'll assume you're referring to Deut. 22:12, which refers to tassels on cloaks, not bedsheets.

A go-to resource for me is the Tyndale commentary series which says this:

The command to wear tassels on the four corners of one’s garment was designed to give Israel something distinctive in her dress. We may compare this with the distinctive dress of the Syrians and Canaanites. A symbolic meaning is given to these tassels in Numbers 15:37–41, namely that they are a reminder to Israel to keep God’s law.

J. A. Thompson, Deuteronomy: An Introduction and Commentary, vol. 5, Tyndale Old Testament Commentaries (Downers Grove, IL: InterVarsity Press, 1974), 258.

The Numbers passage reads:

The LORD said to Moses, “Speak to the Israelites and say to them: ‘Throughout the generations to come you are to make tassels on the corners of your garments, with a blue cord on each tassel. You will have these tassels to look at and so you will remember all the commands of the LORD, that you may obey them and not prostitute yourselves by chasing after the lusts of your own hearts and eyes. Then you will remember to obey all my commands and will be consecrated to your God. I am the LORD your God, who brought you out of Egypt to be your God. I am the LORD your God.

(Nu. 15:37–41; New International Version).

The commentary for the Numbers passage reads:

As a constant reminder of the commandments the Israelites must attach tassels to the corners of their clothing. Deuteronomy 22:12 specifies one at each corner of the square outer garment. Egyptian and Mesopotamian sculptures and paintings from the first and second millennia BC show garments with tassels hanging from them. Whether these tassels were purely decorative (the wearers often appear to be nobles), or whether they had a religious function, perhaps serving as amulets to ward off evil spirits, is uncertain. But whatever their significance in other cultures, their use in Israel is made plain here. Like the mĕzûzôt little boxes containing verses of the law attached to the doorposts, or the tĕpillîn on the wrists, they were designed to promote constant meditation on the law of God (Deut. 6:6–9).

Gordon J. Wenham, Numbers: An Introduction and Commentary, vol. 4, Tyndale Old Testament Commentaries (Downers Grove, IL: InterVarsity Press, 1981), 148–149.

As I quickly look over commentaries of Deuteronomy, it seems 22:12 is usually glossed over with while texts that can skip it over tend to do so. I believe that Nili Sachar Fox ends up addressing why this is the case when they write:

Even ornaments and special bindings such as jewelry, fringes, and headbands are not gender specific. In short, the Bible leaves us with a very limited vision of what constituted Israelite dress and how the dress of males differed from the dress of females. Interestingly, our clearest evidence of biblical Israel’s observance of gendered dress is derived from the prohibition of cross-dressing (Deut 22:5).

"Prohibition of Cross-Dressing in Deuteronomy 22:5," from Mishneh Todah: Studies in Deuteronomy and Its Cultural Environment of Jeffrey H. Tigay, (Winona Lake, Ind: Eisenbrauns, 2009), 65.

While it doesn't seem relevant, I think it does help highlight that the Bible does not even give us a real idea of what people wore day-to-day or why. Texts largely take such matters for granted: Israelites would have known what proper attire was in their culture and perhaps did not give such matters too much thought.

One of the sources Fox recommends, that I unfortunately do not have access to at the moment, is an article by Douglas Edwards that appears in the Anchor Bible Dictionary entitled "Dress and Ornamentation;" I actually have to head to my university tomorrow to return some books so I'll try to remember to take a quick look to see if there's anything of note there.

All of this is a long way to say that people do not seem to understand why fringes/tassels would have been mandated. It seems fair to say that the original authors just assumed their audiences would understand why (in the case of Deuteronomy) or assumed that providing theological justification for cultural practices (as we probably see in the Numbers passage, in my opinion) was sufficient. If this was a practice associated with priestly vestments by neighbouring cultures, my guess would be that this fits into the idea that the Israelites were "a kingdom of priests and a holy nation" (Ex. 19:6). However, the reason given in Numbers could also be considered normative (and is perhaps related): it's a reminder to meditate on what it means to worship Yahweh.