I've used a bunch of websites, but it still doesn't make much sense to me.
I understand that you're saying something that somebody said (1) and that somebody said by exact quote (2)
Can someone explain it?
As soon as you reproduce or paraphrase something somebody else has said or written, you use Konjuktiv I. In correct standard German you always use Konjunktiv I in these situations, no matter if you doubt the content or not. In colloquial speech Indikativ is sometimes used if you don't doubt the content.
Galileo sagte, die Erde sei eine Kugel. (I don't doubt that he was right.)
Mein Ehemann meinte, er trinke nur ein Bier mit seinen Freunden. (Nobody believes that, he will get wasted like always.)
Think this way: if it is a quote, then it is presumed to be accurate. If it is paraphrasing what they said, then there may be some degree of doubt, thus the subjunctive.