When excess of carbon dioxide is passed in lime water, calcium carbonate is converted to calcium bicarbonate which is soluble, hence the milkiness due to calcium carbonate disappears.
$Ca(OH)_{2} +CO_{2} \to$ $\underset{\text{Milkiness}}{ {CaCO_{3}}}$ $?
$CaCO_{3} + H_{2}O +CO_{2} \to$$\underset{\text{soluble}}{ {Ca(HCO_{3})_{2}}}$