If you describe something as specific it signals that you're being precise. Say you ask your friend to meet you at a specific time and place — it means you want to meet her at, say, Ray's Pizza at 3:00 PM, not just anywhere.
Specific also means "detailed and exact." That's the sense of the word that comes into play when a teacher advises you to be as specific as possible. Indeed, specific writing paints a picture. Say your friend Angela lives on a soybean farm in Iowa and takes Claritin for her pollen allergy. That might not be the most interesting fact in the world, but it's a lot more interesting than saying you have a friend who lives in the Midwest and takes medicine for a problem. If the sentence about Angela's pollen allergy is interesting at all, it's because of the specifics.