@Jeanne62
It is against AirBnB policy to take third party bookings - having said that, it happens quite a lot, and many people are happy to take the bookings. Children booking for parents, people booking for weddings, people booking it as a present for someone.
Also, you have let the person into your property. The main thing is that you are not covered by AIrBnB if anything was to happen = having said that, trying to claim anything from AirBnB seems to be close to impossible.
So, you are a bit stuck. Its up to you, but if the person seems fine, etc then use it as a warning for future bookings to double check that the person booking will be staying too.
Good luck