Good question. What we think of as water has more in it than water molecules. Gases from the air can also be dissolved in water. This is what allows fish and other aquatic animals to breathe underwater; there is oxygen dissolved in it. We cant usually see these gases until they form bubbles. If you put cold water in a glass, you will see some bubbles at first as the air that got trapped during pouring escapes. Later, more bubbles will form as the water warms up because warm water holds less air than cold water.

Ice holds even less air than warm water. As ice forms, the water molecules sort of grab onto each other and the gases get forced out of the solution. Some of these gases escape out the surface of the ice cube, but some get trapped. You may have noticed that freezing starts on the outside of the cube. When the surface is frozen, gases cant escape. They get driven into the part thats still liquid. As you may have noticed, thats in the center. So what you are seeing is basically frozen bubbles trapped in the middle of the cube of ice. The surface of the bubbles scatter light, making the ice less clear.

In your ice cube tray, these bubbles may not be too exciting, but in glaciers, they actually give scientists a chance to measure what the atmosphere of the earth was like back when the glaciers (and their bubbles) were formed, sometimes thousands of years ago. Theres a page on this that shows how volcanic eruptions, nuclear accidents, and carbon dioxide in the atmosphere leave their footprints in glacial ice:

This reminds me of one of the most fun questions I have heard, When snow melts, where does the white go? What do you think?