I mentioned in another part of this thread, the payment page as shown on the blog entry does not indicate that my CC# is stored with Stripe, rather from all appearances it appears to be stored with Kickstarter.
Now having read this thread I get that in fact Stripe is storing it, but if I hadn't read the responses to my initial post, I'd just assume Kickstarter was asking to store my CC# which I wouldn't agree to.
The checkbox is whether you want to be able to pay again without retyping it.