勝ち目 literally means a prospect or chance of winning. In the song, the narrator uses it metaphorically when she realizes that, because he loves someone else, she has no chance of winning his heart.
This emotionally charged word is memorable because it frames unrequited love like a hopeless competition, adding a sharp and bittersweet feeling to the lyric.
This is the most painfully polite heartbreak song I know. back number give us a narrator who has been quietly in love with a friend the whole time, while he falls for someone else right next to her. She opens by admitting she already knew: "本当はもう分かってたの" (the truth is, I already knew). Standing beside him, she has no chance, and she says so herself.
What gets me are the small, specific details. He keeps talking to her about the other person, and she keeps listening, because that is still better than not seeing him at all. What she actually wanted to hear was not when the 終電 (shūden, the last train of the night, a very Japanese marker of a late evening out) leaves. She wanted him to notice she cut her hair for today: "似合ってるよって言ってほしかった" (I wanted you to say "it suits you"). That one line hurts more than any big dramatic chorus could.
The chorus is her promise. She only ever wished for his happiness, even if it is not with her, so she will hide her feelings and push him forward "横から背中押すから" (I'll push your back from the side). 背中を押す literally means "to push someone's back", and it is a common Japanese idiom for encouraging someone to go for it. It fits perfectly here, since she has spent the whole song standing at his side. The last line, "誰よりも幸せにしてあげて" (make her happier than anyone), is her asking him to treat the other person well. It is selfless to the point of being a bit much, and that is exactly why it works.
back number is a Japanese rock trio formed in Gunma Prefecture in 2004 by singer and guitarist Iyori Shimizu, with Kazuya Kojima on bass and Hisashi Kurihara on drums.
The band signed to Universal Sigma and made its major-label debut in April 2011 with the single "Hanabira".