Generally, substances are most dense when they are in their solid state. That ice floats on water conclusively proves that water is an exception. We are also aware that lakes freeze from top to bottom. In order for an object to float, it must be less dense that other substances in a mixture. From this, we know that ice is less dense than water, or else lakes would freeze over when layers of ice sink and displace the surrounding water. The reasons for ice floating on water are the unique properties of water and the resultant lower density of ice per unit of volume.
The property that makes water unique is hydrogen bonding. Two hydrogen atoms and one oxygen atom are joined by covalent bonds in one molecule of water. Weaker hydrogen bonds are responsible for bonding among water molecules. The higher ratio of hydrogen to covalent bonds results in the unique expansion at low temperatures. When water reaches below a threshold of four degrees Celsius, the adjustment of the positively charged hydrogen bonds keeps the negatively charged oxygen atoms apart. The resulting "crystal lattice" structure is known as ice.
Ice is estimated to be about 9% less dense than water. Therefore, we have an idea how much more space ice occupies compared to water as well. If one litre of ice is converted to a liquid state, it would produce less than one litre of water. In water, molecules are squeezed together. Since ice is a solid, there is a more defined molecular structure. If one observes molecules of ice, a diamond structure directly results from the adjustment in the hydrogen bonding between adjacent molecules.
It is important to note that the assumption is that we're using similar volumes of ice and water in making this statement. Suppose a block of ice has a higher overall density (not in terms of mass per unit volume) than a significantly smaller volume of water. In such an instance, the ice may not float based on relative densities. If one looks at larger volumes of ice and water, such as icebergs in the ocean, those huge masses of ice still float because of the much greater pressure exerted by the massive volume of water. Ice cubes in your cup of water are just a microcosm of this.
Should someone ask you why ice floats on water, answers like "because it does" or "its common sense" would not suffice. After all, there is a reason for everything - even if we may not know it. To put it very simply, the same quantity of ice would occupy more space than the same quantity of water. All other things being equal, once the solid is less dense than the liquid in a mixture, it will float.