Short Answer:
The civil clerk is in contempt, inhibiting civil rights which are the domain of the federal. The ACLU is 100% right here. Legally, this is the direct equivalent of a civil clerk denying an African American the right to register to vote.
However, the state of Kentucky caused this problem, because does not provide policies and procedures for "conscious objectors" to opt-out, without inhibiting or otherwise delaying the right of those to be married, utilizing other authorities, as do those of states like North Carolina, who foresaw this issue, as even several US Supreme Court Justices warned would happen.
So the right is really not in a good position on this. But the left is ignoring one very basic tenant that has long existed in the US workforce, including for civil servants. Everyone loses.
TL;DR Answer:
Again, some of the dissenting Supreme Court Justices in the prior ruling predicted this would happen. It's the reason why the ruling wasn't 9-0, when it could have very much been so. This is irrespective of the fact that, legally, what this county clerk is doing is not short of denying an African American the right to Voter Registration -- again, legally speaking. But here comes the other side of the same reality ...
I.e., the fact that many states have failed to enact legislation to enable "Conscious Objection" in the case of marrying couples when the county clerk's name is on the license, while they have all sorts of other policies and procedures for similar in many other cases, would be an issue. The clerk was even willing to allow her office to issue marriage licenses, as long as her name is not on them, even though they are from her county. But the state of Kentucky doesn't care, unlike the state of North Carolina that enacted legislation to ensure this.
As a result, the county clerk is in contempt and guilty of Civil Rights violations, as the ACLU correctly asserts. It's not a "gay issue." It's a civics issue, one where the Supreme Court ruled in favor of Civil Rights, as it had to, but several Justices recognized the "Reasonable Accommodation" of "Conscious Objection" would be an issue in some states that failed to implement solutions prior. So the state of Kentucky is to blame for this as much as the clerk of one of their counties.
Good
NY Times Opinion Piece on the Matter:
http://www.nytimes.com/2015/09/07/opinion/we-dont-need-kim-davis-to-be-in-jail.html
Otherwise, understand most of the conservative justices have been deeply disappointed in most of the legal arguments preventing marriage equality, and have been heavily siding with the liberals, as those lawyers have utterly failed to make their cases in terms of civics and public interest. In reading most of the arguments personally, I have to agree. The lawyers on the right-leaning political spectrum seemingly have an IQ of under 80, and fail to put forth arguments that make any sense, legally. Not even ones that attempt to use historical basis.
Including the fact that Anglo-American law is actually based on Judeo-Christian law at points (which is why there are mixed rulings on the display of the Ten Commandments -- whether they are in legal/historical, or religious/non-historical, contexts). Instead, the left utterly smashed their arguments wholesale and absolute, showing the sheer, classic "separate, but not equal," especially considering the dozens, if not even 100+ forms, required to obtain the same level of "union" which quickly becomes a Civil Liberties issue.
The fact that this multi-time divorced county clerk started by denying a couple of 10 years, who patiently waited for it to become legal, and are raising a daughter with down syndrome is just icing on the cake for the left. Anyone on the right who says this is a "jurisdiction" and "States' Rights" issue really needs to step back and realize Civil Rights are one area where the Federal can trump the State, and the US Supreme Court very much has made that determination.
At the same time, those on the left need to realize that the "separation of church and state" doesn't mean any and all "Conscious Objections" are null'n void, and should not be accommodated, not even when "Reasonable Accommodation" is possible, and has been done many times before for civil servants, even county clerks. The recent US Supreme Court rulings on "Reasonable Accommodation," including religious for those of Muslim Faith, very much dictates otherwise.
One of the more recent rulings being
EEOC v. Abercrombie:
http://www.oyez.org/cases/2010-2019/2014/2014_14_86
