切り札 literally means a "trump card," and it refers to a decisive advantage or a final secret weapon kept in reserve. In the song, the narrator calls his obvious lie a 切り札, then tears it up on a full-moon night, suggesting he has finally given up relying on deception.
"空も飛べるはず" means something like "I should be able to fly through the sky," and it's one of the gentlest love songs you'll hear. The narrator starts out as a bit of a mess. He can't shake a childish fever, he's scared of "神様の影" [God's shadow], and he carries "隠したナイフ" [a hidden knife] that doesn't suit him at all. That knife is a great detail: he's trying to look tough, and it just doesn't fit. What gets him through is a silly song, "おどけた歌でなぐさめた" [comforted me with a joking song].
Then he meets someone, and the chorus opens up: "君と出会った奇跡が この胸にあふれてる" [the miracle of meeting you is overflowing in my chest]. Because of that, he figures he can fly now. The second verse is lovely and specific. He tears up the "見えすいた嘘" [see-through lie] he used as his trump card on a full-moon night, and he wakes from deep sleep to the smell of her hair. Even when "ゴミできらめく世界が" [a world sparkling with garbage] rejects the two of them, all he asks is "ずっとそばで笑っていてほしい" [I want you to keep smiling by my side]. It's a modest wish, and that's why it works.
Learner tip: はず means "should" or "is expected to," so 飛べるはず is "should be able to fly." And ～ていてほしい is how you say "I want you to keep doing" something.
Spitz is a Japanese rock band formed in Tokyo in 1987 by four art school students: Masamune Kusano on vocals and guitar, Tetsuya Miwa on lead guitar, Akihiro Tamura on bass and Tatsuo Sakiyama on drums.
They released their first single, "Hibari no Kokoro", in 1991. The single "Robinson" (1995) sold more than 1.6 million copies, "Cherry" (1996) more than a million, and the 1995 album Hachimitsu topped the Oricon chart and won Best Album at the Japan Record Awards.