THE SCIENCE: Softened butter allows small pockets of air to get trapped when creaming it with sugar. This leavens the cookies (aka adds volume) and makes them fluffy. If you use butter that's too soft (such as butter that has been microwaved), it can actually do the opposite and prevent air from getting trapped. And if you were to use cold butter, it would be too rigid.
Long story short: Softening butter is important!