Typical cellular respiration lab with a respirometer -- why is potassium hydroxide (KOH) necessary?
The solid precipitate does not contribute as much to the pressure inside the vial as the gas does. Intuitively, the gaseous molecule contributes to the pressure by collisions that exert force on the vessel (which would push the water out). By contrast, the solid precipitate only contributes to the pressure insofar as it decreases the volume of the vessel, which is negligible in comparison to the loss of kinetic energy per gas molecule.

Basically, just because they both occupy space, doesn't mean they occupy the same amount of space (or rather, they don't contribute equally to the pressure in the vial). Gases tend to occupy more space than liquid or solid phase molecules of identical size, due to their kinetic energy of motion and the force of their collisions on their surroundings. The KOH reacts with the CO2 to remove it from the pool of high energy gas molecules, converting them to low energy molecules in the solid. With the lower number of molecules contributing to the pressure (ideal gas law), the pressure is lowered and the force of the water becomes greater than the force of the gas, and the water enters the respirometer.