The checkbox is whether you want to be able to pay again without retyping it.
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 token can only be used with the private / public key pairs that Stripe provided to them, so even if a hacker got access to Kickstarter's database, they would still need the private/public keys to make use of the tokens.
Also, my expectations are that only whitelisted IPs should be able to access Stripe with the key pairs of Kickstarter.
Basically, the only important information Kickstarter gets from any card are the last 4 digits, whether it's Visa, Master Card, AMEX, etc., and the expiration date.
As others have indicated, this is one more place that my CC# is stored. I have more trust in Amazon's security than in Kickstarter's, although from what I have gathered reading this thread it sounds like the data is stored in Stripe, which I have a tendency to trust based on name recognition.
If the page is to be presented as is though, with no other information, I am not going to store my CC# with yet another .com that hasn't been around that long.
Why not? Any fraudulent activity is your CC company's problem, not yours.
All in all though, this means I will be much less likely to support any Kickstarters.
With Amazon Payments, I did not have to re-enter my CC#. Backing a project became an impulse buy. Click, type PW, done.
Now I have to go get my wallet, type in a bunch of information, which gives me a LOT longer to think about if I really want to back a project or not.
Indeed the primary reason I use Kickstarter over other funding websites is because I don't have to re-enter that payment information so often.