Okay, really, can someone answer this? Why is "di" needed here!? I even had another exercise and the sentence was "Lui dice ciao." I kept trying to figure out how "di" was used here. I knew "He says of no" was obviously wrong, so I put "He answers with no." Then I find out the "di" is not even used. Someone please help us!
I asked my friend from Italy about this mysterious "di", and she explained, most of the time, if you leave out the "di" and just say "Lui dice no", you are directly quoting him as saying "no". But with the "di", it is an indirect quote of what he said. He maybe just said "no", or maybe he gave a long-winded answer and you sum it up as "no".
There is less overlap if the example is not him saying yes/no. If you want to say, "He says to go", you'd say "Lui dice di andare". It's not going to be a direct quote like that, because a sane person would probably not just say the infinitive "to go", full stop. So that sentence will always have "di". On the other hand, with "Lui dice ciao", it will always be a direct quote, so you won't ever use "di". If you wanted to say, "He says hi" as an indirect quote in Italian, you'd use a verb instead, and say "Lui ti saluta".
I hope this helps clear things up!
I am also a Spanish speaker. So, when we say "He says no" we say "El dice no" or also say "El dice que no". The traduction for the last one would be sometging like "He says that no" Sounds weird, yeah, but It actually makes sense for a Spanish speaker. So, with italian "Lui dice di no" the english traduction should be "He says that no"