釘付け literally means "fastened with a nail," but it is commonly used figuratively to mean being unable to look away, as if your attention is pinned in place.
In the song, 今日も釘付けだよ means "I'm transfixed by you again today." This vivid expression captures the singer's helpless fascination with the person she loves, making it a memorable and useful phrase to learn.
This is a crush song from =LOVE, and it is gloriously over the top. The narrator opens with her heart going "キュンてなった" [it went squeeze] and "ぷるんてなった 私のハート" [my heart went all wobbly, like jelly]. Those are sound words: キュン is the little pang you feel when someone cute does something cute, and ぷるん is the jiggle of pudding. Japanese pop lyrics love these, and they are worth learning because people use them in everyday chat.
Then she turns the whole thing into a quiz: "さてここで 問題です" [now, here is a question], who in this huge world thinks about you most? Obviously her. She only watches, wears "ほんの少しだけつけた香水" [just a tiny bit of perfume] and wonders if it works, and even admits "ちょっと重いよね この気持ち" [this feeling is a bit heavy, right?]. 重い (heavy) is real slang for a love that is too intense. There is also a cat-voiced aside, "気付いてないのかにゃ?" [you haven't noticed, meow?], where にゃ is a kitten sound swapped in for a normal ending. Cute or a bit much, depending on your tolerance.
The title line "恋 はじめました" [I've started love] plays on the sign Japanese restaurants hang in summer, 冷やし中華はじめました [we've started serving cold noodles]. The second chorus makes the joke explicit with "夏 はじめました" [summer has started]. Summer is everywhere in the verses: a sunburn mark whose sting "恋に似ていた" [was like love], and an ice cream melting in the sun. By the end she is bossy and greedy in the best way: "この先 私以外 入っちゃダメ 君のハート" [from now on, nobody but me is allowed in your heart], and she has the plan mapped out, a hug a year from now and "エンディングはキスで" [the ending with a kiss].