仕合わせ is a poetic spelling of shiawase, usually written 幸せ for “happiness.” In this song, it suggests more than ordinary happiness: meeting the person you were meant to meet, like two destined threads coming together. Its connection to the verb 仕合う, “to come together” or “to work together,” gives the word a memorable sense of fate and connection.
"Ito" means "thread", and the whole song hangs on one image of weaving. It opens by admitting how little we control: "なぜ めぐり逢うのかを 私たちは なにも知らない" [we know nothing about why we meet], and we never know when, either. Then comes the line everyone remembers: "縦の糸はあなた 横の糸は私" [the vertical thread is you, the horizontal thread is me]. Two people are the warp and the weft, and the cloth they make together "いつか誰かを 暖めうるかもしれない" [might someday keep someone warm].
The middle verse is rougher. "ささくれ" is a hangnail, the small torn bit of skin by your fingernail, and the singer uses it for the leftover sting of days spent lost or days you tripped while chasing a dream. "こんな糸が なんになるの" [what good is a thread like this?] is a genuinely shaky moment, sung while "ふるえてた" [trembling] in the wind. By the end the cloth might even "傷をかばう" [shield someone's wound]. The last line has a nice trick for learners: "仕合わせ" is an old way of writing shiawase, happiness, using the kanji for "doing" and "fitting together", so meeting the thread you were meant to meet is literally what happiness is called.