they send raw credit card numbers, they dont send YOUR credit card number. They create a disposable credit card number, which they send down to the hotel, linked to your credit card. Its like a token in the form of a different credit card number. The hotel doesnt know the difference and the disposable credit card number has a fixed limit which is what you paid for your room. After that it is discarded, I dont know what happens to it afterwards.
Edit: I work for a large hospitality software company, those numbers go thru us before they get to the hotel.