I kept journals for several years. One night I woke up from a dream that was particularly interesting, so I turned on the light, found my journal, and wrote down everything I could remember.
I decided then to keep a journal and a pen next to my bed so that recording additional dreams would be easier. I soon found that I had collected multiple volumes of recorded dreams and auxiliary notes about them.
I don't know whether writing my dreams down made it more likely that I would dream, or more likely to remember my dreams--I know that some people say that it makes you more likely to remember your dreams, but I don't know whether that's true. I just know that the number of dreams I ended up writing down was much greater than I expected based on the number of dreams that I had previously remembered.
Also, I needed to write them down very quickly after waking or I'd forget them. Every minute I waited caused me to lose more details.
That still seems to be true years later. I don't keep a dream journal anymore, but I notice that if I wake from a dream, and I want to remember it, I need to write it down or tell someone about it pretty much right away. If I don't, it'll be lost within a few minutes. I might remember that I had a dream, and I might remember what I felt like when I was having it, but most often nothing more than that.
If I write it down or otherwise communicate it, though, I usually remember the parts that I managed to communicate before the rest disappeared.
Maybe if you keep a journal close to your bed as I once did, you'll get lucky one night and wake up from a dream to write it down. Maybe it will then turn out that those people are right: that writing down dreams makes you more likely to remember additional ones.
I don't know if that's true, but maybe.