There're 3 ways for Google to show the website's short explanation on search results;

1- If the website has a meta description shows it


2- If it doesn't then checks it from dmoz.org
http://www.dmoz.org/search?q=bitcoin.org If the meta description wasn't existed then it probably uses this one;
"Official site offering documentation, forums and the open source client software which permits to send and receive bitcoins. "

3- If none of those exists then it uses the indexed home page's first lines as short description.