Simple. It is persecution of those couples. She is not being persecuted for her beliefs - she is suffering the consequences of not following a court order to perform the duties that she swore an oath to perform. (By the way, the Bible isn't too keen on people forswearing themselves either.) To say that she is 'simply' following a Biblical mandate not to support homosexuality is frankly hypocritical, as she has not bothered to check to see if any of the heterosexual couples she's issued a license to have been previously divorced (which the Bible denounces as adultery).
If you refuse to make a BLT because 'pigs are unclean', you don't get to turn around and make a ham and Swiss on rye without getting called on it.