The Torah itself gives the reason for tzitzit directly: the fringes are to be looked upon so that the people remember and perform all of God's commandments, and not stray after the heart and eyes toward sin, as stated in Bamidbar 15:38-39.
The Gemara in Menachot 43b develops this into a mechanism tied to Shema, teaching that seeing the techelet and white strings leads one to remember another mitzvah dependent on it, namely the recitation of Shema, and further that becoming obligated in this mitzvah obligates a person in all the mitzvot together.
Rashi (Bamidbar 15:39) explains the psychology behind the warning not to stray after the heart and eyes: the heart and eyes act as scouts for the body, with the eye seeing, the heart desiring, and the body then committing the transgression, which is precisely what tzitzit is meant to forestall.
The famous story in Sifrei Bamidbar 115 illustrates this protective function concretely: a man who was scrupulous in the mitzvah of tzitzit was saved from sin with a harlot when his four tzitzit strings appeared before him like four witnesses and stopped him at the critical moment, demonstrating that the reward of the mitzvah is bound up with the mitzvah itself.
Sefer HaChinukh 386 roots the mitzvah in the same verse, explaining that there is no better reminder than carrying one's Master's seal on the clothing one wears constantly, and he cites the idea that the very word tzitzit numerically alludes to the 613 commandments when combined with the eight threads and five knots.
The Ramban (Bamidbar 15:32) situates the mitzvah in its narrative context, explaining that it was juxtaposed to the incident of the wood-gatherer so that the Jewish people would remember all the commandments through it and not forget the Sabbath or any other mitzvah as that man had forgotten; and Midrash Aggadah (Bamidbar 15:36) and Abarbanel on Torah (Bamidbar 15:32) similarly connect the juxtaposition to the gatherer's sin of forgetfulness, with the Abarbanel adding that tzitzit functions alongside brit milah, tefillin, and mezuzah as signs placed on the body and home to keep God constantly in mind.
Rabbeinu Bahya (Bamidbar 15:38), Kitzur Ba'al HaTurim (Bamidbar 15:39), and the Malbim (Bamidbar 15:40) all elaborate on the numerical symbolism, explaining that the five knots correspond to the five books of the Torah and the eight threads guard the limbs prone to sin, so that the sum of threads and knots alludes to the 613 commandments, making the mitzvah equal in weight to the entire Torah; the Recanati (Sh'lach 14) adds that the techelet thread itself recalls the sea, the sky, and the Throne of Glory as a sign placed upon Israel like a king's seal upon his servants.
Later halachic codifiers translate this purpose into practical guidance: the Rambam (Mishneh Torah, Fringes 3:11-12) rules that although one is not strictly obligated to acquire a four-cornered garment, a person should always try to be wrapped in one so as to fulfill this mitzvah, because Scripture made it equal to and dependent on all other mitzvot; Shulchan Arukh (Orach Chayim 24:1) and Arukh HaShulchan (Orach Chaim 24:2) likewise urge wearing a tallit katan all day so that a person remembers the mitzvot at every moment, through the five knots recalling the five books and the four corners recalling remembrance in every direction; and Mishnah Berurah (24:1) and Shulchan Arukh (Orach Chayim 24:3) add that one should look at the tzitzit when donning them and reciting the blessing, since sight leads to remembrance and remembrance leads to action.
Among the responsa on the sheet, Igros Moshe (OC 4:4) affirms that although there is no strict obligation to wear a tzitzit-requiring garment, the accepted practice is binding precisely because tzitzit serves multiple purposes together — remembrance, leading to action, protection from sin, and receiving the Divine Presence.
Az Nidberu (7:90) holds that the primary purpose of the mitzvah is to wear tzitzit constantly as a reminder of the mitzvot and as a safeguard against following the impulses of the heart and the misleading sights of the eyes.
Tzitz Eliezer (8:4) argues that wearing a tallit katan throughout the day is integral to the mitzvah itself, since continual sight of the tzitzit helps a person remember all the mitzvot and avoid following the heart and eyes, and on this basis he urges every person to obtain and wear a four-cornered garment all day.
Further sources reinforce and extend this remembrance theme in various registers: Shaarei Halacha uMinhag (Orach Chaim 13) states that God commanded tzitzit so that seeing them leads to remembering and performing all the mitzvot, hence a tallit katan should be worn visibly over the clothing; Kitzur Shulchan Arukh (17:7) and Kaf HaChayim (Orach Chayim 8:31) describe the practice of holding and gazing at the tzitzit during the recitation of the relevant verses in Shema precisely for this reason; and Kitzur Shulchan Arukh Yalkut Yosef (Orach Chaim 8:10) and Rav Mordechai Eliyahu (Maamar Mordechai Limot HaChol 7:3) record the custom of verbalizing this intention before wrapping oneself in the tallit.
Several sources deepen the theme on a more esoteric or homiletical level without abandoning the remembrance-of-mitzvot framework: Tanya (Likkutei Amarim 41:5) describes donning tzitzit as drawing down the light of the Ein Sof clothed in Torah, tzitzit, and tefillin; Chiddushei Aggadot (Shabbat 155) explains tzitzit as placing the body under God's service alongside tefillin placing the soul under His service; Chiddushei Aggadot (Menachot 29) explains that the fringes extending outward symbolize bringing all the mitzvot into practical action; Meshekh Chokhmah (Shelakh 56) reads the unwoven threads as teaching that even a person's free choices require divine assistance; Gevurot Hashem 45 ties the remembrance of the mitzvot in the tzitzit passage to the remembrance of the Shemot; the Alshich and Alshekh on Torah (Bamidbar 15:2) stress that tzitzit works only when worn with conscious intention to remember, not as mere rote habit; Daas Torah (Rav Yerucham Levovitz) frames tzitzit as surrounding and safeguarding the person entirely; Likkutei Sichos 2, Shelach presents tzitzit as the tool enabling a Jew to remain spiritually elevated while engaged in worldly affairs; and Sefat Emet (Shelach 22:3-4), Sefat Emet (Shelach 8:5), Imrei Binah (שער הציצית יב,ב), and Imrei Binah (שער הציצית ח,ב) develop kabbalistic explanations of how the garment and its threads draw down holiness and inner wholeness upon the wearer.
Finally, Teshuvot HaRadbaz (Volume 6, 2260) and Be'er Sheva 52 address the practical weight the mitzvah carries relative to other mitzvot — the Radbaz ruling that tzitzit should be donned before tefillin because it is considered equal to all the mitzvot, and the Be'er Sheva explaining the wording of the blessing despite the mitzvah not being strictly obligatory — both reflecting the same understanding that tzitzit exists to keep all of God's commandments before a person's eyes and heart.