So, let's say I'm gay, and I want to marry my boyfriend Gary. We walk into a cake shop hand in hand, and order a wedding cake. The owner is some homophobic religious zealot who tells us he doesn't like gays and we are going to hell. Who cares, we don't believe in hell anyway, but, he better bake the damn cake.
Why? Because as a commercial business, he relies on a business license in order to stay open and charge for his services. And to keep that license, he is prohibited from discriminating against someone based on race, sex, religion, or sexual preference. He actually does have a choice, he can choose not to bake the cake, in which case I would file a complant with the local licensing board, the BBB, and every local blog or rating service I could find, as well as a lawsuit to shut him down and sue for damages [emotional distress, etc, whatever my attorney can come up with]. He would still be able to bake cakes, just not to legally sell them as a licensed business. Still free to pursue his artistic muse, so to speak, but unable to profit from it.