I see you are learning spanish too, so I tell you, if I can help you whit this, that is the equivalent of saying "él dice que sí" in spanish.
Let's see: is not the same "lui dice sì" and "lui dice di sì"... the first is to affirm he says "yes"... the second is, for example, to affirm that he wants something, for example:
"Qualcuno vuole la birra?" (Does someone want the beer?)
"Lui dice di sí. Vuole la birra" (He says yes, that he wants the beer).
It's very common to confim a question or something, but always derived of a previous queston or decisionmaking...
"Lui è una brava persona?" -Is he a good person?
"Io dico di no" - I say no (it's my opinion, and I think he isn't a good person)
I try to help as a spanish speaker. I hope I was as clear as possible :)
From my Italian wife: “Lui dice che si” isn’t correct in Italian. Either “lui dice di si” o “lui dice si”. They are both essentially the same but the first one can refer as an answer to a specific question whereas the other could be used as well to just state a fact (e.g. you don’t know what question was asked, you just know the man is saying yes. But you can really use the second in any situation).
I think the reason Elena18 asked this question is because in later topics you get sentences such as "lei dice che lo vuole" ("she says she wants him" or more litterally "she says that she wants him" and the che is required or duo marks it as wrong)... I understand "dice di sì" and "dice sì" but how does the che fit into the example I gave