On the campaign page, click "Manage your pledge," then "Update payment method," then "Add a new card" to enter your new details and confirm. The same option is available under "Manage pledge" in the Kickstarter app.
A few things to know: adding or removing a card in your Kickstarter account settings doesn't automatically update the payment method on a live pledge — you need to update it on the pledge itself. If you have more than one pending pledge, each needs to be updated separately. Payment details for pledges are handled entirely by Kickstarter — we don't have access to them on our end.