Go to Manage Your Content And Devices, on your Kindle device.
(Ask for help if you cannot find that... it isn't easy on some older Kindles.)
Then, choose the Settings
then choose the Edit Payment Method (in the Digital Payment Methods section perhaps)
then choose the (presumably expired?/outdated) credit card already entered there
then choose Edit (or choose Edit my card then choose the card, in some cases, or chose Continue perhaps...)
then, correct or approve each part of the credit card's details, until you get to the expiration date, which probably is what needs updating to the new expiration date, if you have received a new card last month.
Then save those changes.
Let me know how that goes, esp. if there's any trouble with the process... I will help you further as needed.