Bonds don't pay interest, they pay a "coupon".
The value of the coupon of a corporate bond is set by the corporation which issues the bond. They are free to pick any value they want (including zero). But if they set the coupon too high, they will pay more capital acquisition cost than they have to. If they set the coupon too low, they won't find any investors willing to buy their bonds. So they will usually set it at the lowest value they still expect to be high enough to attract investors.
While the central bank interest rate doesn't directly affect the coupons of private and public bonds, it does so indirectly. Bonds compete with other forms of financial investment. When the interest rate is high, then bonds might have to increase their coupons in order to still be an attractive option for investors compared to just parking their money on a bank account.