Kernel Embedding of Distributions

This seems very interesting, but the overview of this article tests my mathematical knowledge, and I can hardly read the rest without a healthy dose of intuitive reasoning to infer the unknown terms. As far as I can tell, this is a method of statistical analysis which does not require assumptions about the structure of the data in order to perform statistical operations, including comparing data. Conventional information theoretic methods require modeling the predictability (or entropy) of data and then performing statistical operations on the basis of those models, which may be errant, oversimplified or difficult to determine for complex data sets. Instead, the data are represented with an arbitrary number of dimensions in a way that generalizes Euclidean space, and then spatial operations can be performed on the data.

That's as far as I can understand and I'm afraid that there are mistakes even in my simplistic summary. Can someone explain it better?

It exceeds my mathematical knowledge, and I gave up trying to read it. My inference was "use of some kind of transformed space when modelling distribution" but the benefit/result versus a direct analysis was absolutely a mystery.

I would still love an explanation of the practical benefits of its use (preferably without the use of numbers or formulae) and IMHO this is what Wikipedia should present. If anyone here is are able to write that description, please do.