After some skimming of the paper and googling of terms, I'm going to take a whack at an ELI5 (note I have no special domain knowledge here): Some fields such as GIS or CAD are given a polygon and need to represent the interior area as a 'mesh' or set of triangles in order to perform certain operations on that interior. The current approach to building a mesh gives an approximation of that area where the method in the paper gives a mathematically exact representation, which is a 'big deal' where accuracy counts. Corrections welcome.

Fry: Hey, professor. What are you teaching this semester?

Prof. Farnsworth: Same thing I teach every semester: The Mathematics of Quantum Neutrino Fields. I made up the title so that no student would dare take it.

High order meshing is becoming increasing important in Computational Fluid Dynamics since high order numerics requires the boundaries to be more accurately represented. Would be interesting to know if this method could be extended to generate quadrilaterals.

As someone with only a basic knowledge of FEA, why would a quadrilateral mesh ever make more sense than triangular?

There are multiple reasons but they all can be application specific.

First reason is that quadrilaterals can be mapped to a square and so you can use "Tensor product" elements which is just a fancy way of saying that you can uncouple the equations to be 1D for each direction. So you can think of your problem in 1D and easily extend them to 2D by using quadrilaterals.

Second is somewhat related and is sparsity. The matrices generated for quadrilaterals would be sparser and hence your performance will be higher.

Third is accuracy. Quadrilaterals are regarded to be more accurate however I am not sure there are proofs or conclusive studies on this, more a rule of thumb.