惹かれてく comes from 惹かれる, meaning "to be drawn to" or "to be attracted," with 〜てく showing that the feeling gradually develops.
In the song, the singer says he is becoming mysteriously drawn to someone: 少し不思議なくらい惹かれてくのさ. The spelling with 惹, meaning "to attract or captivate," gives this romantic phrase a poetic nuance beyond ordinary physical attraction.
This song runs on a calendar. Each season gets its own promise to the person he loves: in spring, "花をあげよう" (I'll give you flowers), in summer, "きれいな花火をしよう" (let's set off pretty fireworks), in autumn he'll chase a dream, and in winter, "君に寄り添っていよう" (I'll stay close beside you). The fireworks line is a very Japanese summer picture: hanabi, literally "fire flowers", are the big event of a summer night there, so spring's flowers turning into summer's fire flowers is a neat little wordplay a translation would flatten. The hook ties it together in a mix of Japanese and English, "君へ all year around falling in love" (to you, falling in love all year round).
The second half gets bigger and sillier, in a good way. He opens "ふさいだきりの扉" (a door that has stayed shut), then shouts "炎ともして殼を壊そう!" (let's light a flame and break out of the shell!), and by the bridge he's waving goodbye to physics: "さよなら重力 僕は自由だ!" (goodbye, gravity, I'm free!). The flying image "高く羽根を広げ" (spreading my wings high) leads to the real point, "何処へでも行けるさ 君といたなら" (I can go anywhere, as long as I'm with you). My favourite line is the blunt "愛だけじゃ足りない" (love alone isn't enough), which sounds odd in a love song until you notice he keeps adding fireworks, flames and wings on top. The English bits even give a tiny origin story: "It was a fine winter day / Remember when I met you". It's sugary, yes, but it knows it.