There are two problems with Kim Davis's refusal to issue licenses to same-sex marriage couples. One is her illogical stand on religious freedom and the other is a total misunderstanding of what she represents.
I celebrate religious freedom, people have the right to believe whatever nonsense they want, but Ms. Davis is doing just the opposite. She is proclaiming that only her very narrow religious viewpoint counts. How about the religious liberty of the the same-sex couples whose religious beliefs are clearly contrary to those of Ms. Davis. There are plenty of Protestant denominations that have no problem with same-sex marriage. If this were truly about religious freedom both sides rights would have to be honored. As it is, the Davis group only wants their own tiny religious view represented. Should she prevail, a non-Christian religious zealot could deny marriage licenses to any Christian who walked through the door. Her position is the antithesis of religious freedom.
Secondly, Ms., Davis has confused herself with the office. When she signs (or more likely rubber stamps) a marriage license she is not personally approving anything. She is acting as an officer of the county and legal system. It's not about her personally, it's about her acting as a representative of the state and county. She has personalized her job. It would be like a library circulation clerk refusing to check out Huckleberry Finn because the word "nigger" appears in the book. The clerk's function is to check out books, not to foist his own perspective on others. Ms. Davis, a thrice divorced woman - which would seem to indicate she doesn't value a marriage license much anyway,-- doesn't get that. The marriage license is a "civil" contract saying the "state" is giving permission to marry. There is no religious context or approval involved.
We are very fortunate in this country not to have a religious test to hold office. But we are also fortunate in that we expect office holders not to let their personal faith determine their public actions. Just as we as citizens recognize that the money we pay in taxes may be used to support things in which we do not personally believe (much to the consternation of some theocratically-oriented religious groups) such as pacifists supporting the military, etc. Kim Davis was told by the governor of her state to follow the law; if she could not according to her conscience, she was free to resign, much as a conscientious objector can perform alternate service rather than fight in the military. She chose instead to make a public issue of her refusal.
My cynical mind draws an inevitable conclusion: this is a play to make a lot of money. By becoming a public "martyr" for her "cause," she will inevitably attract considerable amounts of money, just as did the pizza joint in Indiana, to be followed by lucrative speaking engagements.
For a dispassionate account of events, the best I've seen is at http://blog.constitutioncenter.org/2015/09/constitution-check-is-freedom-of-conscience-a-shield-against-public-duty/