The supply is lower now. The demand is the same. The prices must go up from the people competing for the smaller number of rooms.
This is also why it caused rents to rise.
Except it hasn't. Hotel prices are the same today in NY as they were when Airbnb was going full blast. Again, NYC has always had excess hotel inventory. There is no shortage that would effect pricing. You can check right now on Expedia and see that prices are roughly the same as last year and the year before that.
The supply is lower now. The demand is the same. The prices must go up from the people competing for the smaller number of rooms. This is also why it caused rents to rise.