The Cartesian diver demonstrates the concept of density as a function of mass and volume. In the bottle's uncompressed state, the medicine dropper floats because there is not enough mass in the volume of the medicine dropper to make it sink (i.e. its density isn't great enough). It is mostly air, and this air is much less massive than the water surrounding it. The mass of the dropper and the small amount of water inside of it doesn't contribute to the total mass (and, as a result, the density) enough to make the diver sink.
When the walls of the bottle are squeezed, the pressure increases inside the bottle. This added pressure decreases the volume of the gas in the bottle, including the gas in the medicine dropper. This is Boyle's Law. When the volume of the gas inside the medicine dropper decreases, the space left behind is filled with water (which is more massive than air). As a result, the mass inside the medicine dropper has increased substantially, but the volume has remained the same. More mass in the same amount of volume yields a greater density. The density of the diver increases to the point where it is now greater than the density of the water surrounding it. The Cartesian diver sinks.