I'm not sure there is such a tidy link between a democracy and a legal system. The Normans (proto-France) were the first (after the collapse of the Roman Empire) to introduce a complex code of laws to Europe and brought those to England during their invasions. Most of our legal system today has its roots in those Norman laws, in fact many of the terms still used today came from that time. All of that was long before any democracy in the region.
But, to get to the heart of your question... a good, healthy democracy would need at its base a healthy system of law, where citizens have a means to obtain recompense for wrongs done to them (such as replacing damaged property), and to obtain relief from ongoing harassment (such as the arrest of someone doing injury to them). Even if these mechanisms are in place, if the citizenry is not aware of them, what good would any of it be? If you owned a car and were unaware that it belonged to you, you would never drive it, and it would be useless at best -- a nuisance at worst.
Although the citizens do not need to know they ins-and-outs (there are lawyers for that), they do need to know what options are available to them, at least. Without law, any civilization (democracy or otherwise) would collapse in anarchy.
Citizens in a democracy need to know about their legal system for several reasons. For instance, they need to know what their rights are and how to elect the best representatives who will not abuse their power.