She did not take the position that she should be allowed to deny gays any good or service. She only objected to participating in a gay wedding. Committed Christians believe in a doctrine of vocation. They believe that their work is a form of ministry. Through their work, they can share the gospel and glorify God. Because committed Christians believe marriage is a relationship created and ordained by God Himself to be between a man and woman, they believe they cannot provide goods and services to a marital union that would run counter to that which God ordains. Christian merchants do not see themselves as passive participants in a transaction, but active in a ministry. Their work cannot be separated from their faith.
The government saw it otherwise and forced the florist to perform the work or be punished.
Similar situations have come up in Colorado, Oregon, New Mexico and other states with florists, photographers, bakers and others. None of them denied all goods and services to gay. They just declined to provide them with goods and services for a gay wedding because of their faith.
A number of states have sought to ensure Christians cannot be compelled by the state to violate their consciences. The laws are being badly mischaracterized as anti-gay. Christians are being compared to Bull Connor for trying to honor their God. The state is picking sides in matters of conscience. Instead of living and letting others live, tolerance has become a one-way street. Those who seek to dissent or opt out are made to care whether they want to or not.