Two things happen when you microwave the soap: the soap gets heated, which makes it softer, and the air and water trapped inside the soap gets heated, causing the water to vaporize and the air to expand. These expanding gases cause the soap to expand and become a foam (popping popcorn is a similar process). Microwaving Ivory™ soap changes its appearance does not cause a chemical reaction, only a physical change. Charles' Law states the volume of a gas increases with its temperature. The microwaves add energy to the soap, water, and air molecules, causing them to move faster and further away from each other making the soap puff up. Other brands of soap don't contain as much air and just melt in the microwave.
I have done this experiment with my summer science camp kids and they loved it! The ivory soap expands like a cloud because of the air and moisture it contains. Other soaps won't do this because they don't have as much air in them.