The original use of L-systems is to model the fractal-like forms that appear in plant growth. Johan Knutzen gives a nice overview with pictures in Generating Climbing Plants Using L-Systems.
More recent applications include musical compositions with a self-similar structure, as described by Stellios Manousakis in Musical L-systems. (This musical rendering of L-sytems has been called Growing Music.)
In computer science, L-systems are used to generate the space-filling fractal curve that maps IP addresses to computers (Hilbert curve).