切り刻み comes from 切り刻む, meaning “to cut into pieces” or “to shred.” It is a vivid, dramatic expression that goes beyond the ordinary verb 切る (to cut).
In the song, the “scissors of the era” are said to cut the speaker’s sail into pieces, creating a powerful metaphor for time and changing circumstances tearing life apart. The lyrics then promise to stitch those pieces back together, giving the word an especially memorable emotional impact.
Sakanaction's "Years" is built around one picture: a sail. The singer looks ahead and sees "待ち受けてる時代のハサミは" (the scissors of the era waiting for us), and he admits they will probably cut this sail up: "切り刻みバラバラにするけど" (they'll slice it to pieces, but...). That "but" is the whole song. His answer comes straight after: "また繋ぎ合わせるから" (because I'll stitch it back together again).
It opens small and windy. "風が吹くのを見逃さないように" (so I don't miss the wind when it blows), he left "乱れた髪さえそのままにしてたんだ" (even my messy hair just as it was). I love that detail: he won't even fix his hair in case he misses the moment. Then the request that the sail metaphor was leading to: "思い出しても許してくれるかい?" (will you forgive me if I remember you?). Notice 時代 (jidai) here means an era or the times, not a clock time, and the ending かい is a soft, slightly old-fashioned way of asking a question.
The last lines play a neat grammar game. "僕の中の変わらないこと" (the unchanging thing inside me) becomes "僕の中で変わらないこと" (the thing that doesn't change within me). Swapping の for で shifts it from a thing that belongs to him to something happening inside him, and it's a great pair to study side by side. He settles on "多分これが変わらないことのひとつ" (maybe this is one of the things that won't change), and the "this" is left for you to fill in: probably the memory of the person he's singing to. The Floating Points remix stretches all of this out, so the repeated "Years, years" gets plenty of room.