As water cools down, it is able to hold less and less material in solution. Eventually the water freezes, and most of the solute will be left behind in the unfrozen water under the ice layer. (Sea ice, though, is white rather than clear because it traps solid salt within itself.)

Lakes don't freeze all the way down, so minerals in the water move down as the water on top freezes, and the lake ice is clear. Sidewalk ice is often frozen from relatively pure water (rain or melted snow).

But ice in your freezer doesn't have anywhere to shed its minerals, and it freezes faster than lake ice anyway. So, like sea ice, the minerals which were originally dissolved in the water are trapped as very small crystals within the ice cubes. If you make ice with purified water, you normally get clear ice cubes.

Incidentally, in freezers with automatic defrosters, ice will eventually sublime (auto-defrost works by subliming away the frost from within the freezer). If you leave ice in for a week or two, you end up with much smaller cubes coated with the mineral dust once dissolved in the water.