Okay, that’s confusing, because the option is available on pages when editing them.
Right, I understand it’s a bit confusing. As I mentioned earlier, if you add a featured image to a page, although it isn’t output on the page itself, it’ll be picked up as the thumbnail image when sharing the page on social media like Facebook.
Here’s an example using this page on the demo:

Nevertheless, let’s say you can’t display featured images on pages. Then how do you get images with the beautiful vignette (or whatever it is that gives it the fancy “edging”) as shown in the demo:
The front page of the demo shows posts, not pages. ??
Pages and posts are different. Posts are displayed on the blog in reverse chronological order, and that’s what you’re seeing here: https://buttondemo.wordpress.com/
If you add featured images to your posts they’ll be displayed on the blog with the border cutout.
Static pages, on the other hand, won’t display featured images unless you make the theme tweak I explained above.
You can learn more about the difference between posts and pages in WordPress.
Let me know if this explains things a bit more!