Why do they hate us?
OK, here's the thing.
A pluralistic society, in effect, is like the breakfast cereal aisle at Publix. Man, so many choices, good and bad. Cap'n Crunch. Granola. Old School Oat Meal. Total. Except, instead of cereal, we have every flavor of philosophical belief system. Islam. Fundamentalist Christianity. UFO Conspiracy Theory. Atheism. Kabbala for the Stars. Leather Fetishism. Believe or not. Put whatever you like in your bowl. It's all there -- minus any government endorsement.
Basically, the principle implies that your belief system is either a private delusion or irrelevant. You're free to believe what you like -- in the privacy of your own whatever. You can't impose your beliefs on others. God says don't drink; don't eat cows; avoid transfusions; whatever. You're free to follow whatever internalized set of rules you like. God, as far as the state is concerned, is your invisible friend, like Harvey the Rabbit. If he doesn't tell you to hurt self or others, it's no problem. (The state may draw the line at, say, multiple wives or making bong hits a communion ceremony. You're free to obey, but only up to a point.) You can't claim to be God's spokesperson. You're not free to tell others to obey. (See David Koresh.) Obey God. Don't play God. That, for some, is the problem.
The separation of church and state directly threatens the authority of any earthly organization (and the holy authorities running them) claiming to be on a mission from God. It strikes at the base of their power: their claim to divine authority.
Aside from propping up dictatorial choads with our petrodollars, that's why they hate us.