The principle is based on density: weight per unit volume.

Consider an object that sinks in pure water. What you know about it is that its density is greater than the density of water. Can you measure or estimate its actual density? (No, unless you want to get into sinking velocity.) Now consider an object that floats. Yes, you know its density is less than that of water, but what is the density? If the object is a perfect cube and you can measure the depth of submergence and the side length, that ratio is equal to the object's density divided by the water's density, hence you can calculate the object's true density.

In Archimedes' crown displacement, he may have chosen to use a dense saline solution rather than pure water, so that the crown just barely floated and he could calculate the crown's metal composite density.