The site you've referenced doesn't actually have the video showing in place of the featured image, it just a screenshot of the video eg -
the actual video is embedded in the post so you click on that image, it takes you to the post and then the videos within the content etc.
I think Martin wrote the above as you said you're using the viper video plugin in your original post.
If you want to use videos instead within the post, go to create new post in the top right click on screen options and enable video -
You should then see all the options for video in the post screen below the main area where you insert the post content -
Add you link/embed code etc and publish.