差し伸べる literally means “to stretch or extend,” but it is often used for offering help or support to someone in need. In the song, the singer promises to keep reaching out to the lonely person, making this a warm and memorable expression with emotional depth.
This is a love song about being right next to someone and still not reaching them. The chorus opens with "まだ 届かなくて もどかしくて" (it still doesn't reach you, and it's so frustrating), followed by "こんなに傍にいるのに" (even though I'm this close to you). He kisses her, he holds her, and the answer is the same every time: "心に届かない" (it doesn't reach your heart). The repetition of "抱き締めても 抱き締めても" (even if I hold you, even if I hold you) sounds a little desperate, and I think that's the point.
What I like is how specific the picture of her gets. She "人波の中 孤独に溺れたんだね" (drowned in loneliness in the crowd), she smiles while hiding her sadness on a sleepless night under "月のシルエット" (the moon's silhouette), and she has old scars that still hurt. The line "誰かに逢いたいのに 一人になりたかったんだね" (you wanted to see someone, but you also wanted to be alone) is the most honest moment in the song. His answer is stubborn: "望まないとしても 差し伸べ続けるよ" (even if you don't want it, I'll keep reaching out to you).
Learner tip: "イタくて" is 痛くて (it hurts, and...) written in katakana for emphasis. Also watch the "〜ても 〜ても" pattern, meaning "even if... even if...", which carries the whole chorus. It ends on a promise: "温かな 手を掴んで 離さない" (I'll grab your warm hand and won't let go).