ACLU base its arguments on the fact that this law regards marriage between same-sex marriage as a "second-tier marriage" and therefore qualifies it as "unconstitutional on its face". Note that the law should go into effect on July 1 as Mississippi Governor Phil Bryant signed the bill into law in April.
This law aims to provide protection based on religion, but not only because the text also provides protection based on moral conviction. It obviously opposed to marriage between same-sex couples, but also to the recognition of gender, and allow discrimination by businesses and by government officials, such as those who issue marriage licenses.
Photo from aclu-ms