As Jacob Milgrom points out, there are numerous ancient Near Eastern reliefs in which certain people are depicted with tassels on the hems of their garments. These hems are displayed as scalloped, cut like an umbrella, where arches meet at points around the circumference. That meeting point is where the tassels project, and the tassels are in fact extensions of the embroidery of the hem, rather than strings added as attachments. The tassels would either hang by themselves or have flowers or bells embroidered at the tips. 
The below relief found at the Medinet Habu mortuary temple of Ramesses III depicts what might have been the original tzitzit.
|Egyptian Relief c. 1190 BCE in the burial complex of Ramesses III|
The relief shows representatives of various peoples conquered by Ramesses III. Among them, second to left, is a Semite with tassels, and at the far right is a Philistine with tassels. Each of the tassels hang from a kanaf, i.e. the downward, "wing-like" projections of the hem.
Which begs a couple of questions: If this is the sort of tzitzit the Torah had in mind, clearly this was a fashion style of the day. Why should it be made into a "commandment"? And if indeed other peoples wore tzitzit, why would the Torah say: "See it and remember all the commandments and do them" (Num 15:39)? I would expect the "reminder" object to be distinct to Israel, something which specifically evoked the commandments, as opposed to tassels which the Philistines and other peoples also wore.
Milgrom speaks about hems and tassels being highly significant in Ancient Near Eastern cultures:
The more important the individual, the more elaborate and the more ornate was the embroidery on the hem of his or her outer robe. The tassel must be understood as an extension of such a hem.He goes on to cite Akkadian texts where people's hems being cut off signify the removal of a part of the person's essence. This applied to exorcism, wherein cutting off the hem was done in order to purge an evil spirit. Likewise, divorces were effected by the husband cutting off the hem of his wife's robe. A prophet would "sign" his report to the king by enclosing a lock of his hair and a piece of his hem, or an impression of his hem would be made on a clay tablet.
In the Bible too, David famously cuts off the hem (kanaf) of Saul's robe. (I Sam 24:4) This was significant not simply because it meant that David could have killed Saul, but also because the hem itself symbolized Saul's position of authority - which was "cut off." And indeed Saul later responds, "Now I know that you will become king." (v. 20)
An additional feature of the Israelite tassel is the explicit command to add a string dyed with tekhelet (sky-blue). Tekhelet dye, being prohibitively expensive at the time, was primarily reserved for the noble classes, royalty, and priests.
Being commanded to wear decorative tassels containing tekhelet on the hem of one's garment might thus be understood to convey the Israelite self-concept of being a "kingdom of priests" (Ex 19:6). It was part of the Israelite democratization of religion and priesthood, whereby holiness and elevated conduct were not reserved for a small and elite priestly class, but things to which every Israelite individual was enjoined to aspire. A person would gaze at the tzitzit and be reminded that whatever their socioeconomic status, they were a part of a larger project, a higher calling, a covenant.
Which in effect answers the question as to why a specific Israelite reminder - the tzitzit - should be something worn in other societies. To use another example, if I have a wedding band on, the fact that other couples or people of other religions wear them does not diminish its personal significance to me. Just the opposite - the fact that a wedding band has a universal meaning enhances its personal significance. Likewise, the meaning in ancient Israel of such a hem + tassle + tekhelet configuration would have been readily recognized and understood. It symbolized elevated status and significance, and also responsibility.
Kings, prophets and other people of authority were not simply "privileged" classes - they had a great sense of weight upon them as being responsible for the welfare of the people, a responsibility which they saw as stemming from the divine. For the Israelites, the status of being "holy" is contingent upon accepting the responsibility of the commandments:
לְמַעַן תִּזְכְּרוּ וַעֲשִׂיתֶם אֶת כָּל מִצְוֹתָי וִהְיִיתֶם קְדֹשִׁים לֵא-לֹהֵיכֶם
In order that you remember and fulfill all my commandments, so that you will be holy to your God. (Num 15:40)
Yes, observing the mitzvot should bring personal enjoyment and fulfillment. But part of the meaning we derive from the mitzvot is the greater calling that they evoke, where it's not about what we "get" out of them, in this world or any other. And it's certainly not the perverse notion that the mitzvot - or being Jewish - makes us inherently "higher" or "better" or "holier" than anyone else. It's about embracing our responsibility to lead by example with ethical scrupulousness, interpersonal grace, and to elevate the world around us with compassion, creativity and joy. That level of deep caring and refinement is what it means to be a "priest." It is a weight of responsibility that we lovingly and eagerly accept, and for which tzitzit is meant to serve as a reminder.
1. See Baruch Levine, Anchor Yale Bible, Numbers Vol. 1, p. 388; R. E. Friedman, Commentary on the Torah, p. 479; J. Milgrom, Of Hems and Tassels, Biblical Archaology Review, Vol. IX, No 3, May/June 1983.
2. Brown-Driver-Briggs, edge, extremity; Hebrew Aramaic Lexicon, skirt of a garment.
3. J. Milgrom, Of Hems and Tassels.