DOMPurify is a common HTML sanitizer library that helps block untrusted HTML code into entering the page. Most importantly, it helps prevent XSS.
Mutation XSS is the browser/library selectively altering the HTML in order to produce XSS where it did not exist before. In this case, the author used a quirk of the specification: forms cannot be nested inside of each other.
There are three types of namespaces (in a DOM tree): HTML, SVG and MathML. These three different namespaces are parsed separate ways, but do have odd integration points.
With these different namespaces and integration points, it was possible to do mutation XSS that was not even an XSS before! This is because a form tag gets stripped, resulting in the namespace of a tag to change.
Overall, good write up for the XSS with good background. In the future, I think I'll be looking at the specs more :) This guy uses references the HTML specification quite a bit!