How do Chladni plates make it possible to visualize sound?

More specifically, you are able to see the standing waves along the nodal lines of the plate. The sand moves away from the antinodes, where the amplitude of the standing wave is maximum, and toward the nodal lines, where the amplitude is minimum, forming patterns known as Chladni figures.

What do Chladni patterns show?

Chladni patterns show the geometry of the different types of vibration of violin plates. This site has an introductory explanation of modes of vibration and a library of photographs of the Chladni patterns of the bellies and backplates of two different violins (one mass-produced and one hand-made).