I'm going to have to disagree with the "illegible" part, since there is a five-pound book entitled The Notebooks of Leonardo da Vinci sitting on my desk. Obviously, someone was able to read his handwriting.
As for the "reverse handwriting," there may be a less-than-exciting motive behind it. All evidence – particularly the direction in which he cross-hatched to shade his drawings - points towards Leonardo's having been left-handed.
Let me explain why this is significant. When you are a "lefty" (as I am) and work with a wet medium such as paint or ink, or even a dry medium like charcoal or pencil, it's nearly impossible to avoid dragging the outside of your left hand through anything you've put to paper or canvas. Unless you work from right to left. This may sound crazy if you're right-handed (and 90% of all humans are), but it is relatively easy for us southpaws to work this way, and also to read standard Western text upside-down and/or from right to left.
Leonardo point being: "They" told me in grade school that Leonardo used "mirror writing," and wasn't that too deliciously mysterious? Didn't buy that explanation then - while busy mucking up my No. 2 penciling in a right-handed, spiral-bound copybook, all the time sweating over lost neatness points - and haven't since. As a fellow left-handed person, I assumed he wanted to get his observations written down as expeditiously as possible, and didn't want to worry about smearing his ink. (Before you email me, I would like to publicly acknowledge that my theory here is boring. Practical, and also plausible, but boring.)
The image above is that of one page (11 r.) from the Leicester Codex (probable dates 1506-1510), a collection of 18 double sheets of paper upon which Leonardo wrote thousands of lines of his observations on water and the science of hydraulics. Every single line is "backwards." Leonardo also sketched some 300 illustrations throughout, usually within the right-hand margins.