The whole song hangs on one invented word. 晴る (haru) is built from 晴れ (hare, clear weather) but it is pronounced exactly like 春 (haru, spring). Yorushika swaps between the two spellings all the way through, so every time you hear haru it means spring and clearing skies at once. Watch the written lyrics and you can see the switch: 「晴るのせい」 (it's the clearing's fault) in one chorus turns into 「春のせい」 (it's spring's fault) in the last one.
The singer is watching someone, 貴方 (anata, you), who seems far away. 「貴方の目はビイドロ」 (your eyes are biidoro) is a lovely detail: ビイドロ is an old word for glass, borrowed from Portuguese vidro centuries ago, so their eyes are clear and a little fragile. The singer keeps asking 「何が悲しいのだろうか」 (I wonder what makes you sad) and never gets an answer.
Then the weather does the talking. The chorus shouts commands at the sky: 「晴れに晴れ 花よ咲け」 (clear up and clear up, flowers, bloom!) and later 「泣きに泣け 空よ泣け」 (cry and cry, sky, go ahead and cry). Blaming the tears on the rain, 「泣いて雨のせい」 (crying, and it's the rain's fault), is such a gentle way of letting someone be sad without saying so. My favourite line is 「降り頻る雨でさえ 雲の上では晴る」 (even pouring rain is clear sky above the clouds).
Grammar tip: that よ after a noun, as in 花よ, 空よ and 歌よ, is a poetic way of calling out to something, a bit like "O flowers". It sounds old and songlike, which suits a track that ends with 「さぁこの歌よ凪げ!」 (come on, song, go calm now!).