You're staying in someone's home or apartment instead of a hotel. Sometimes AirBNB is cheaper than a hotel but not always. Sometimes it is the only option if all the hotels are booked. Or, in the case of California's hottest markets, the only way to find a place to stay, hence the backyard tents for rent.
They are renting it out for a day or week. You don't know what they are offering except the listing.
They may or may not be there while you are. In NYC, they must be there to legally rent it out.