Yeah, he is using WordPress.com, says so at the bottom of the page. He's just using the custom domain option.
http://en.support.wordpress.com/domains/
Some people like to have a welcome page and have their blog on a different page. I'm not sure how easy it is to do that on WordPress.com (or if it's even possible), but it's easily doable on a self-hosted install.
It's basically duplicating the index page, adding a comment saying that you want it to be known as "Whatever", in this case "Blog", and then making a page with that template. Then you change the front page to "static" in the settings and choose the welcome page.
You cannot do this with WordPress.com, not even with the paid plans.
For self-hosted, you want a domain, and then a web hosting provider. Here are the requirements to run WordPress:
https://wordpress.org/about/requirements/
Quote:
PHP version 5.2.4 or greater (recommended: PHP 5.4 or greater)
MySQL version 5.0 or greater (recommended: MySQL 5.5 or greater)
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And then mod_rewrite is recommended.
Some hosts will have something like CPanel or DirectAdmin that lets you install it with one click. 