Lime water is a saturated solution of calcium hydroxide. When calcium hydroxide reacts with carbon dioxide, calcium carbonate is precipitated.
Ca(OH)2 + CO2 --> CaCO3 + H2O
If an excessive amount of carbon dioxide is passed through the lime water, the solution becomes clear again as the calcium carbonate that was initially formed reacts with CO2 to form Ca(HCO3)2 which is colorless.
When a person runs on a treadmill, the body cannot produce energy by aerobic respiration as it requires a large amount of oxygen. Instead, anaerobic respiration is used which is inefficient as compared to aerobic respiration and produces only one-tenth the energy.
Anaerobic respiration does not produce CO2, lactic acid is the only product. This has to be eliminated from the muscles and that requires heavy breathing to increase the intake of oxygen necessary for this process.
After running on a treadmill, if one were to exhale through a straw into lime water immediately, it would not turn cloudy.