The reason salt increases the density of water is that the mass of the resulting saltwater solution increases, while the volume is unaffected. The density of a substance is given as the mass (m) per its volume (V). The equation is density = m/V.
To study density of different substances, it is convenient to compare two containers, where one container has plain water, and the other container has salt, or sodium chloride and water. In this experiment, both containers have equal amounts of water. The container that holds the water has the mass of water only. However, the saltwater mixture contains the added mass of sodium and chloride ions. This causes this mixture to have a greater mass while the volume changes by a slight amount. Using the density formula, it is evident that the salt-water mixture with a greater mass and a volume change that is negligible will have a greater density.