As described in the Kobold Entry of Races of the Dragon, the Draconic script was actually invented by Kobolds who wanted/needed to write down what their Dragon masters taught them, because their lifespans and thus their memories are so insignificant compared to true Dragons. This is, for as far as I know, the only official information regarding the usage of the Draconic script (at least in non-setting 3.5 sources).
Aside from that, the other answers given here are very likely. Dragons that can shape-shift can indeed turn into a creature that can operate a quill and start writing, or they set someone else to write for them.
Keep in mind that all Dragons, even good-natured ones, are very vain and arrogant. I can well imagine one of the less driven ones seek some sort of immortality by writing an auto-biography. And I can also imagine a quest where PC’s have to save slaves from an evil Red Dragon, only to find out that the Dragon has captured a whole village worth of authors and is dictating his own life-story to them so they can write it down FOR him, allowing him to choose the best entry to publish (and possibly eat the rest).
However, at the same time, I imagine that most Dragons don’t actually write all that much. Depending on the setting you’re in, Kobolds came to be aeons after Dragons (although the aforementioned RoD seems to imply otherwise). If Dragons were able to hold out without a script for this long, surely they can do the same even now that the script is here. Some may see it as a useful tool, and use it as such. Others may see it as an insult that those lowly Kobolds tried to capture their perfect tongue in crude and childish drawings.