How to show featured images and small text description in search availability results ?

If you need to show the description and the images in the search results like in this search availability demo.

Then the text  in search  results is getting from "Excerpt” section  of  post,  where you have inserted the booking form 
and the image - its "Favorite image" of this post.

In search  result form  at the Booking > Settings > Search page you will be need to  use these shortcodes:

[booking_featured_image] - featured image, taken from the featured image associated with the post,
[booking_info] - booking info, taken from the  excerpt associated with the post

In case if you are using pages with booking forms and not the posts.
You need to install  some other plugins,  like "page-excerpt" plugin, which will  add the excerpt section to the pages,  as well.
The "Featured Image" by  default is supporting in Pages,  as well. But this functionality  must  support by your theme.
So please contact  support of your theme,  about  adding ability to  use "Featured Images" in your pages.