The option price you see listed on your quote page is equal to intrinsic plus extrinsic value. In order for the market maker to figure out how much extrinsic value there should be he uses a theoretical option pricing formula, and one of the inputs in that formula is an interest rate. Subsequently, each market maker may have slightly different credit and debit interest rates which are used to figure out their actual "cost of carry".
In other words "cost of carry" is one input in a formula to calculate extrinsic value.
In other words "cost of carry" is one input in a formula to calculate extrinsic value.