It does this by connecting to Twitter on the web and screenscraping the data, rather than by using the Twitter API.
You can install TwitRSS.me by following the instructions in the INSTALL.md file.
The main reason for doing so is that the caching limits (one query every 45 minutes) is proving restrictive.
You can see a bunch of TwitRSS.me blog posts on Charlie Harvey's site. They mostly deal with me being increasingly freaked out about the amount of traffic that is turning up.
The code for TwitRSS.me is released under the GNU GPL licence version 2 (See the LICENSE file) and is copyright of the contributors — Charlie Harvey and Daniel Perelman at time of writing.
The website content is released under CC-BY-SA — Charlie Harvey and James Lu at time of writing.