It's the referrer field. When you click on a Washington Post link on the Google News web page, and your browser packages up an http request package to send off to the Washington Post web server, asking for the desired web page, one of the things your computer puts in that first outgoing packet is the "referrer", which is the URL of the Google News page holding that link you just clicked on.
Then when the Washington Post's web server gets your request for that page, a fraction of a second later, the code running in that web server can look at a field in your request called the "referrer" field, and give your request special treatment. For instance, if it that fields says you clicked through from Google, it could skip the registration. Apparently, it is doing just this.
That The Proxomitron tool that is mentioned, on the web site USS Clueless - Washingon Post Registration linked to off the one you provided, is apparently a tool that lets you lie about this referrer field, and set it to, say, http://www.google.com when asking for a Washington Post page, so that you can take advantage of their special treatment for requests coming from (referrer field value) Google to bypass the registration.
As clear as mud, no?