Why is the carbon dioxide we exhale transparent?
Essentially, because the carbon dioxide sublimates from solid (dry ice) to gas at a very low temperature (roughly −78 °C at 1 atm), it causes water vapour in the air to condense, causing a visible fog. Thus what you are seeing is not carbon dioxide, but rather water. 

When we exhale and it is reasonably warm, the carbon dioxide expelled is roughly body temperature (36–37.5 °C) and so we do not see any fog forming.

What happens when we exhale in cold weather is a bit different. Our breath contains a high percentage of water, higher than the percentage of water that can stay in gaseous form at a low temperature. Thus the ‘extra’ water condenses into droplets.

For reference, here is the phase diagram of $\ce{CO2}$:



And here is the phase diagram of $\ce{H2O}$:



And here is the saturation fraction of water at various temperatures at 1 atm (sea level):