First, it's about not having a federal religion. The states had their own state religions, and this said there'd be no federal one. There were state-level religions for a while in some places. They weren't unconstitutional, except as redefined a century later.
Second, it's about the freedom to exercise your religion. Now, this gets tricky in all sorts of ways.
For instance, most laws are rooted in morality. Things like traffic signs are utilitarian, but the whole death-penalty and abortion mess is a question of morality. Anti-sodomy laws were such. Anti-miscegenation laws. The civil roots movement and abolitionism. To a great extent, many social-welfare laws. And many aspects of morality have their roots, the way they're made coherent and passed from generation to generation as human societies have done since we've had culture, in religion.
When cities needed to be organized and provide for the urban populations they produced granaries. They had rulers who collected all the grain and made it available; there were communal bakeries run by the state. And this was couched in religious terminology and acts because it was easy to codify, there were people whose job it was to ensure that everybody knew the rules, peer pressure kicked in, and it was long-term stable. (This is Mesopotamian stuff.)
To argue that morality and laws should have no commonality, that religion and morality are unrelated, is to stick one's head deep in the sand. We can establish systems of morality, but typically they are rooted in some assumption. That that assumption tends to be something taught early on to the establisher of that system, often religious in its history, and unquestioned.
That said, there are two ways that religious folk want liberty. The one is to have laws that express their morality. In this, they're like atheists, who also have some sort of system of morality and want laws that are in keeping with that morality. That those laws are couched in non-religious language is not horribly relevant here. "You can't argue for the death penalty on religious grounds. I can argue on moral grounds." They mean about the same thing, but it's damned handy to try to rule out a priori your opponent on some sort of technicality.
The other is to not be imposed on. Now, many who argue against religion-induced laws understand this quite well. But it cuts both ways. Think of the laws as not "religion-inspired" but as involving either some forced omission or some forced commission.
Many laws prohibit bad, morally objectionable or simply dangerous behavior. That can be rape or it can be going the wrong way on a one-way street.
Other laws impose a behavior: You must drive on the right, you must pay your taxes, you must take proper care of your dog. Some are avoidable: don't have a car, don't have a dog, screw two of those requirements.
We "get" prohibiting behavior. But a lot of people take offense at the idea of prohibiting behavior they like, whether it's same-sex intercourse or cock fighting. Still, it's what most laws are, at least most that I know of that affect individuals just being individuals. Even then, there are often religious exemptions to some of the prohibitions: you could have wine for communion in prohibition, or use psilocybins when otherwise banned.
The others are more problematic. A lot of people don't like laws that stipulate how you take care of your dog or cat, raise your kid, etc. They force you to not avoid behaviors, but to put your will and effort into doing things you don't want to do. It's like forcing somebody to say the Pledge of Allegiance or participate in a parade.
Now look at the sets of laws that people like to merge into "they're rooted in religion." It's one thing to say, "No, gay sex is legal. Stay out." That's disposing of a prohibition. But telling somebody, "Okay, now we've changed the rules and you must be sure to oblige that guy who wants to have sex with you" would be reprehensible. You're being forced to do something you don't want to do. These are different in principle.
It's why blue laws could survive but laws requiring church attendance couldn't. One is "you can't" and the other is "you must."
The contraception-insurance requirement, the "you must serve gay couples" is the latter kind of law. It forces people to do what they think is wrong. We have enough trouble with jaywalking laws, with school rules that are deemed culturally inappropriate. The freedom to do something has to be built on the freedom not to do something. And not to extort people into going against their conscience: "Either you violate your conscience or you are denied your livelihood." We used to like the idea of freedom of conscience; it was a progressive touchstone for quite a while.
Now we use the idea of civil rights to say, "You not only cannot do certain things, but you must do what we say." Many civil rights advocates are Puritan in the way they want to enforce their moral principles, and take their ability to practice what they want as paramount over others. Ultimately, rights that with maximum range of application are to be found in freedom to do things not forcing others to do things for you. As soon as you start treating others like serfs, you've moved away from rights not into privileges but into entitlement. Lords were entitled to corvee.
If you don't separate out these two classes of laws, there's no way to have anything but a self-righteous and self-justifying dialog.
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