"I believe that what they said... (was... what?) = je crois que ce qu'ils disait... (était... quoi ?)
With your sentence, we are expecting a development, like ... was true / ... était vrai, because you have introduced a relative pronoun "that/que" which calls for a full clause.
Let's break it down. Je crois (I believe) ce que (that which) ils disaient (they said <or> they were saying). There is no verb tied to "ce que" that would allow one to get "this is" what they said from it. I.e., there is no "is" that can be derived from anything in the original sentence.
"You said you have a red bicycle. I believe what you said." vs "You said the proletariat must rise up and destroy their capitalist oppressors. I believe in what you said!" . You "believe in" statements of faith, justice, morality... philosophical stuff. You "believe" anything you consider/trust to be true, no matter how mundane.