Most times, cancellations fees are not paid, they are charged. This is why even if you have to pay on check-out, you are asked for a credit card number, although some hotels reservations do not ask for a card, in which case they have no way to charge you.
For most reservations, when you press the Cancel button on Booking.com, any applicable fee will be charged to your card immediately. If instead, you do nothing and not show up, either Booking.com or the hotel will charge you something which could be the value of the entire stay or the first night, again this depends on the hotel policy.
Once the charge is on your credit card, then you will have to pay the balance unless you dispute the transaction with your credit card company. Since your booking was subject to certain rules, there is little chance of them agreeing in your favor.
As per another question on this site, if you did not provide a credit card, it seems that you generally won't be charged but a hotel could send you a bill the the cancellation fee if they were so inclined.