These are designed to allow for people to opt out of following certain laws because doing so violates their conscience. It's not entirely new - a conscience clause was originally proposed to be part of the Bill of Rights, and conscientious objectors have been able to resist the draft - but it has become more and more common for religious groups to use these to try to avoid following laws they find objectionable.
Not surprisingly, many involved sexual morays - specifically birth control and homosexuality.
The author argues that the impact of these laws are spreading more and more into vast areas of public policy. She calls this conscience creep:
. . . what’s really wrong with conscience clauses? We all have consciences and laws that exist to protect us from being forced to violate our religious and ethical principles should be welcome on the left and right. The problem isn’t conscience clause legislation so much as what we might call conscience creep: a slow but systematic effort to use religious conscience claims to sidestep laws that should apply to everyone. Recalibrating who can express a right of conscience (i.e do corporations have a conscience?) and what the limits of that conscience might be, may well be the next front in the religious liberty wars being waged in courts around the country.
In the current craze to deploy conscience arguments to scuttle unpopular provisions of the Affordable Care Act, birth control is just the tip if the iceberg.
The explosion in conscience claims was kicked off by the decision in Roe v. Wade, which led to a national wave of legislation protecting those with religious objections from participating in abortions. States and the federal government rushed to promulgate conscience clauses for health care workers seeking to be exempt from providing or assisting in abortions. Passed in 1973, the Church Amendment provided that “receipt of federal funds did not require an individual or institution to perform sterilizations or abortions if it would be contrary to ... religious beliefs or moral convictions.” Since then, most states and the federal government have passed laws allowing health care providers to opt out of procedures that offend their religious convictions.
But it hasn’t stopped at health care providers, and the list of objectors now encompasses pharmacists and ambulance drivers, cashiers in supermarkets and business owners who object to same-sex marriage. Last year, for instance, a prison guard withheld an abortion pill from a prisoner who’d been raped on the grounds that it violated her personal religious beliefs. And it hasn’t stopped at abortion, birth control, or sterilization, but may include activities like counseling rape victims or teaching AIDS patients about clean needles.
