The idea is to take the Processing reference page and to try to implement most of it, as faithfully as possible, in another language: JavaScript, Ruby, Python, Scala, Groovy, Clojure, and so on.
Ideally, one can take any of the official sample and run it unchanged. In practice, some languages are too different of Java, so the code needs to be adapted, but still uses background(), ellipse(), etc.

The interest? Well...
Processing.js was basically a proof of concept, to show it was possible. I think it was also made for fun, like most of these implementations, and it has the advantage of being more lightweight (and pervasive) than a Java application.
Other implementations, I suppose, are made for similar reasons, and because people like the simple and effective API, but prefer to use their favorite language instead.