I think that it is "He said with shining eyes", but what a word "させ"? Why "させ+ながら"? Why not "しながら"?
キラキラする is intransitive "shine". eg 「瞳がキラキラする」"his eyes shine"
You use causative キラキラさせる "make something shine" here. eg 「瞳をキラキラさせる」"make his eyes shine"
You can split this into two actions:
「彼は瞳をキラキラさせた。」"He made his eyes shine" +「彼は言った。」"He said."
So the sentence literally means: "He said, while making his eyes shine." Hence the translation "He said with shining eyes".
(When combining two actions with ながら, you should use the same subject for both verbs. i.e you can't use two different subjects for the two actions, eg 「彼は、瞳がキラキラしながら言った。」 is incorrect.)
させる is the causative form of verb する. In this sentence, it implies that 彼 made some kind of effort (conscious or not) to make his eyes shine (possibly to emphasize what he had to say).