A bond is a promise to pay a coupon and, at the end, the face amount. If you buy that promise and hold it, the coupons do not change because the market changed its mind the next day.
The price does. Suppose a new bond of the same maturity offers a higher yield. The old bond, with its smaller coupon, has to cheapen until a buyer is indifferent between the two. That cheapening is the price decline. Nothing mystical happened to the issuer. The comparison changed.