The cloudiness is a mass of tiny air bubbles, air that was dissolved in the water and expelled when the water froze. You can see the individual bubbles through a magnifying glass.
There is always some air dissolved in any water that has been exposed to, well, the air. For this, the world’s fish are truly grateful. They are particularly grateful for the fact that, even though air is only about 21 percent oxygen, oxygen dissolves in water twice as readily as the other 79 percent, which is mostly nitrogen.
When water freezes, the loosely moving water molecules settle down into rigid positions. In doing so, they squeeze out the dissolved air molecules, because there is simply no room for them. When the water begins to freeze, the outer portions freeze first because they are in the best position to have the heat sucked out of them.
As the dissolved air molecules are squeezed out, they become trapped within the encroaching casing of ice. The air molecules are forced closer and closer together as the growing wall of freezing water closes in on them.
Eventually, they are packed so close together that they congregate into bubbles. And there they remain, trapped when the interior water finally freezes.