蜃気楼 (しんきろう, shinkirō) means "mirage", an optical illusion that makes something distant appear real. It is a vivid and poetic word that is much less common than everyday terms for dreams or memories.
In the song, the singer chases a 蜃気楼 of the person he has lost, capturing how unreachable and unreal his memories of her feel. The image beautifully connects longing, summer heat, and the illusion of being able to see her again.
This is a breakup song told from the far side of summer. The singer has a trick for getting over love: every time it ends, he tells himself "あの人はもう死んだ人" (that person is already dead to me). It stops working when the weather turns. The new season is "意地悪" (mean, spiteful), only half warm, and that half warmth is enough to bring her back: "君がよみがえる" (you come back to life).
The best image in the song is the sunflower. Right now he calls himself "モノクロのヒマワリ" (a black-and-white sunflower), a flower that still turns toward the sun and toward her, but has lost its color. Back then he was "虹色のヒマワリ" (a rainbow-colored sunflower), and "太陽は君だった" (the sun was you). By the end he wonders "未来の僕は何色のヒマワリ?" (what color sunflower will I be in the future?). It is a neat way to track a whole relationship through one flower.
The English chorus begs for one last kiss from a "Falling Angel", and the Japanese lines in between get darker and more desperate: "愛で汚して欲しい" (I want you to stain me with love) so he can be "生まれ変わる" (reborn). He never gets it. The song closes with him quietly saying "「さよなら」と呟いた" (I murmured "goodbye").
Learner tip: "欲しい" after a て-form verb, as in "満たして欲しい" (I want you to fill me up), is how you say you want someone else to do something.