It's written in classical Japanese.
なれ is 已然形, which is equivalent to 仮定形 (-ba conditional stem) in modern Japanese, of an auxiliary verb for assertion なり. 已然形 is used there because of 係り結び of the particle こそ. You can think it as copula だ/である.
なり[助動]の意味 - 国語辞書 - goo辞書
係り結び - Wikipedia
せ is 未然形 (imperfective form) of す, which is equivalent to する, and る is 連体形(attributive form) of an auxiliary verb for perfect tense り. You can think it as している or した.
すの意味 - 古文辞書 - Weblio古語辞典
り[助動]の意味 - 国語辞書 - goo辞書
輝ける is also "輝く + り", thus, the verse means as follows in modern Japanese (and therefore the given translation is not correct).
ことばは沈黙に
光は闇に
生は死の中にこそ
あるものである
飛翔しているタカが
虚空でこそ
輝いているように