Yeah, you're right, we do leave out the action. However, it is very important that you either use a direct quote, or just a concept "He is an excellent Writer of Shakespearean Style Poetry" What do you mean by "He is an excellent writer of" or What do you mean by "Shakespearean" In writing, you would the example words. In speech, it is a matter of style. If you feel confident, you should try to copy the intonation and rythm of the phrase. Some might use Air quotes (making a gesture) but this is not always popular. These styles would help to set your speech aside from the speech you are quoting. It would help the listener to know what is part of your question, and what is not. It would still sound natural to say 'What do you mean by the term: "Shakespearean style poetry"

You can say "What do you mean by that?" or "Why do you say that?" Without repeating the entire question. Example Johnny says "You are so stingy." Mary says "What do you mean by that?" Another Example Sam says "You're so stingy." Mary says "Why do you say that?" Another Example John says "I don't like that teacher, he's mean." Mary says "What do you mean by that? I think he's nice." Another Example John says "I don't like that teacher, he's mean." Mary says "Why do you say that? I think he's nice."