Tzitzis illuminate the three soul-levels

איתא ציצית על שם הסתכלות מציץ מן החרכים וכו' ג"פ ציצית שיש ג' מיני הסתכלות והשגות לאיש ישראל בבחינות נפש רוח נשמה.

It is taught that "tzitzis" derives from the term for gazing, as in "peering (meitzitz) through the lattices" (Shir HaShirim 2:9), etc.; the word "tzitzis" appears three times — for there are three kinds of gazing and perception (hasagos) for a Jew, in the aspects of nefesh, ruach, and neshamah.

The Sefas Emes connects tzitzis to "gazing," and notes its threefold repetition corresponds to the three levels of the soul — nefesh, ruach, neshamah — each with its own mode of spiritual perception.

וע"ז רומז הגמ' בטעם תכלת שים דומה לרקיע ורקיע דומה לכסה"כ ובאמת הנפש נאסר בתוך הגוף כמ"ש הוציאה ממסגר נפשי ונתן הקב"ה לנו מצות לסייע להאיר אל הנפש מחשכות הגוף והטבע.

And to this the Gemara alludes in the reason for techeiles (the blue thread): the sea resembles the sky, and the sky resembles the Throne of Glory (Kisei HaKavod); and in truth the nefesh is imprisoned within the body, as it says, "Bring my soul out of confinement" (Tehillim 142:8) — and HaKadosh Baruch Hu gave us mitzvos to assist in illuminating the nefesh out of the darkness of the body and of nature.

The blue of techeiles leads the eye upward from sea to sky to the Divine Throne, a ladder of ascent; the soul, trapped in the body's darkness, is given mitzvos as the means to draw light into it.

וע"י הציצית מאירין הנר"ן באדם כמו שמבקשין ע"י מצות ציצית תנצל נפשי ורוחי ונשמתי כו'.

And through the tzitzis the nefesh, ruach, and neshamah (nuh-rah-nun) are illuminated within a person, as we request by means of the mitzvah of tzitzis: "may my nefesh, my ruach, and my neshamah be saved," etc.

The mitzvah of tzitzis kindles all three soul-levels, which is why the accompanying prayer asks that nefesh, ruach, and neshamah all be redeemed.

והיה לכם לציצית והי' לשון שמחה שצריך איש ישראל להיות משתוקק תמיד לזכור ולהתדבק בשורש הנפש ורוח ונשמה.

"And it shall be for you as tzitzis" (Bamidbar 15:39) — "and it shall be (v'hayah)" is an expression of joy, for a Jew must constantly yearn to remember and to cling (deveikus) to the root of his nefesh, ruach, and neshamah.

Chazal teach that "v'hayah" connotes simchah; the joy of tzitzis is the Jew's perpetual longing to remember and attach himself to the supernal root of his soul.

ואז מצות ציצית מסייע לו לזכירה זו:

And then the mitzvah of tzitzis assists him in this remembrance.

When a person cultivates that yearning, the tzitzis become a constant aid in remembering his soul's Divine source.

Summary: The word tzitzis, meaning "to gaze," appears three times to correspond to the three levels of the soul — nefesh, ruach, and neshamah — each with its own spiritual perception. The techeiles draws the eye upward from sea to sky to the Throne of Glory, for the soul is imprisoned in the body and the mitzvos were given to illuminate it. Through tzitzis all three soul-levels are lit, and the mitzvah — an "expression of joy" — aids the Jew in his constant yearning to remember and cling to the Divine root of his soul.