Hello,
Please check you have no javascript error on your site. Javascript errors can disable functionality of our component.
You are probably logged in as a customer or you allowed booking for guests (non registered customers)?
When I checked you site, I can't see any reservation type there. Maybe you have not set it correctly. I can only recommend you to check manual. Booking can be difficult for new customers, because of many options and specific logic of the component. But it allows you to create many booking systems.
New booking installation should also contain sample data. If you didn't install them, you can also check our demosite on:
http://demo.tramin.artio.cz/joomla25/en/demo/book-it
We also prepared demo accounts for our customers so you are able to check our backend. More info on link above.