掌返した (てのひらかえした, tenohira kaeshita) comes from the image of turning over one's palm. As an idiom, it describes someone abruptly changing their attitude or switching sides, which fits the song's critics who suddenly praised the dream they had mocked.
The singer keeps a list of everyone who looked down on him. "アイツは俺をバカにした" (That guy made a fool of me), "夢見た理想を笑った" (He laughed at the ideals I dreamed of), and taught him how boring reality is, "大人 のリズムで" (in the rhythm of the grown-up world). The worst entry is his own: "ふざけてなぜか誤魔化した" (I joked around and somehow covered it up), and he ends up angry at himself for laughing along. Each blow is a "Heavy Damage", and the first verse closes on a promise: "今に見てやがれ!" (Just you wait and see!).
The chorus turns the damage into fuel. "今日の不可能は" (What's impossible today) "明日可能になる" (will become possible tomorrow), and "最後に笑えばいい" (all that matters is laughing in the end). By the second verse it has happened: "ヤツらは掌返した" (They did a complete about-face) and "笑った夢 を讃えた" (praised the dream they'd laughed at). But success has a cost. He admits "心に悪魔を育てた" (I raised a devil in my heart), and the song ends with him handing the hurt back: "そこに跪け!" (Kneel down right there!), "お前も喰らえ!" (You take the hit too!).