Why are some parts of this ice block cloudy and other parts clear?

As @pr1268 explained, tap water is not pure: it contains dissolved gases (basically air) and dissolved minerals.

However, I do not think stratification causes this phenomenon: I think that as long as water remains unfrozen, the dissolved gas concentration remains approximately constant throughout the solution (basically equal to the saturation concentration). Here is an alternative explanation.

When ice crystals start to form, they naturally tend to exclude the impurities; so the impurities are "squeezed out" into the part of the water that remains liquid. Eventually the impurity concentration exceeds the saturation threshold, and they start to precipitate out of the solution. But by this time, the ice has already formed an airtight enclosure around the liquid water which prevents gas bubbles from escaping to the surface; and the mineral crystals stay no matter what.

For this reason, on a big lake where only a small portion of the total water freezes, you can get crystal clear ice (unless the water surface has snow, slush or other impurities to begin with).

The cloudiness is caused by dissolved air bubbles. Plus, tap water is notoriously rich in mineral particles (giving nucleation sites).

Seeing how you placed the tap water in the freezer, the cold temperature kept the dissolved gases and minerals in solution, and only partial stratification occurred before the water froze. The expanding ice¹ "magnified" the dissolved gas bubbles, causing the cloudy appearance. The clear section is that portion of the water from which the gas bubbles and solute had precipitated (sinking in the container, so to speak).

¹ Water is one of the very few substances that expands when transitioning from a liquid to a solid state.