When you have added Amazon Pay to your subscription checkout, your customer will first be prompted to log in using their Amazon credentials. After being authenticated, their securely stored address and payment information will be presented for them to confirm the billing agreement for the subscription plan.
When the subscription is confirmed, the billing agreement (and relevant Amazon info) is stored for the customer with Recurly. On the following subscription renewal cycle, the same billing agreement consented to by your customer will be used to bill the subscription automatically.
Customers can modify their billing agreement (such as changing payment method or canceling) either from Amazon or through Recurly. On the Recurly side, you can update the billing info via the API.