English Translation
They said yes.
Isn’t it supposed to be “Ils ont dit que oui”? This is something I am often corrected on in classes - e.g., it is “peut-être que non”, and not “peut-être non”
You like to scratch into subtle grammatical questions!
Ils ont dit que oui is correct and suggests a citation.
Ils ont dit oui is correct and only conveys the meaning of what they said.
The verb dire (and répondre) can be used in that form with a few key words: oui, non, bonjour, bonsoir, adieu, au revoir, merci, ouf, zut and… merde.
French grammar being duly infernal, note that dire que bonjour / merci / ouf / zut, etc… are not correct, but dire que oui/non/si is …
You’re welcome 
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Ha! Makes perfect sense that verbs like “dire” and “répondre” have more complex rules.
Grateful for the depth of the explanation, and for your patience! 
I’m by no means an expert in french, but I would have expected it to be the other way around, namely that “que” introduces the unquoted meaning?
That would match with what @paris2020 said, that “peut-être non” is wrong - because you can’t quote “peut-être” but you can quote “ils”, and with what @Anxos said, that “dire que bonjour” is wrong - because “bonjour” is not a meaning or fact and can only be quoted.
I don’t mean to correct anything, just asking to clarify because I’m curious and confused.
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