Lime water is a solution of calcium hydroxide (Ca(OH)2). When CO2 gas is passed through lime water, the following reaction occurs.
Ca(OH)2 + CO2 → CaCO3↓ + H2O
(white precipitate)
The solution of lime water turns milky due to the formation of calcium carbonate, which is a white precipitate insoluble in water.
If CO2 gas is passed in excess amounts, then the above reaction occurs further.
CaCO3↓ + H2O + CO2 → Ca(HCO3)2
(soluble in water)
The product formed is calcium hydrogen carbonate, which is soluble in water. Hence the milky solution becomes a clear solution.