Let's put this in terms everyone can understand.
Kim Davis has all rights to believe what she wants to believe. But she cannot "exercise" her religion as an employee of the Government of the Commonwealth of Kentucky, it is unconstitutional! FACT.
The Constitution forbids government from endorsing religion or engaging in any behavior that does so. When you do what Kim Davis has done, you have breached the Constitution. What I see people doing is clinging onto one part of the First Amendment while ignoring the other. "Government shall make no law respecting an establishment of religion." FACT.
The Government also has rights under the constitution. You cannot make government endorse religious beliefs. It has a constitutional right not to endorse religion in its administration, and is obligated not to. FACT.
When you swear an oath of office which involves doing so impartially, you must do so impartially. If you have been elected to office, you have been elected by and on the good faith of your constituents. When you violate that good faith, when you stop fulfilling your duties for which you have been elected, you breach the oath you swore to them to uphold the law and that office. FACT.
By all merit, Kim Davis not allowing her deputies to issue licenses was her imposing her religious beliefs on her deputies, which is unconstitutional and a breach of oath. FACT.
The law (Section 228 of the Kentucky Constitution) that Davis was sworn to obey was not crafted with any caveats, it was crafted to ensure obedience. She broke that law. FACT.
There is no law which says anyone has a right to marry, but there are laws forbidding government from endorsing the marital practices of one religion over others. FACT.
The 14th Amendment comes into play here. You cannot make laws which benefit one group while disparaging another. People born and naturalized in the United States shall be subject to its jurisdiction and that of the states wherein they reside. When you deny licenses to gay couples you violate the 14th Amendment. In general if you deny straight couples licenses, you are doing likewise. FACT.