Not "book" - just click the words "check in" in the date area:
And it expands to this:
and you use the top arrows to navigate to the month you want and the available dates are dark in color.
This is a pretty common navigational type for website design and calendar function so I'm not sure how many people are being discouraged. My guess is that most people know what dates they want to travel and start with that and are served up all the properties that are available during the desired time frame. People who come directly to your landing page may be looking to see what's available and would need to navigate in this way.