They seem to run a browser on the server and let the user interact with it to choose DOM elements to monitor for changes. I would have just taken a screenshot (perhaps with http://urlbox.io/ or perhaps with CutyCapt) and allowed the user to draw boxes over the screenshot. Then repeatedly screenshot the site and whenever the contents of that box changes, alert the user.
The Sieve method has the advantage that you are able to tell the user what the new text is. The screenshot method is significantly simpler.
EDIT: So, in startup terms, the screenshot method could be the MVP :). Provided, of course, that it is considered "viable" not to know the text. If I were a user I would consider it viable - it is still a vast improvement over having to check the site manually.