They are solving the linear system Ax=b, written in Julia/Matlab usually as A\b, specifically for a fixed size 5x5 matrix. The Grassmann algebras can give a way of representing the matrix and vectors, and a comparison is being made to the representation (and solver) given by StaticArrays.jl, which just uses typical matrices and arrays but optimized for small sizes. No real motivation to use Grassmann algebras if your only concern is generally solving a linear system, other than this method seeming to be faster, which mostly (imo) points to the StaticArrays.jl method needing to be optimized.

Edit: This post also does not mention numerical stability of the Grassmann.jl method, which is a real concern in practice, even for such small systems.

This is a pretty neat trick, they implemented exterior algebra see https://en.wikipedia.org/wiki/Exterior_algebra

You see this kind of math in textbooks focused on computer graphics.

Do you have a Physics background by any chance? Anyway, I believe the module might be named after the mathematician, not the number system called after the same mathematician.

Regardless the name Grassman is tied to exterior algebras (and therefore geometric algebras), which are implemented by this library and which can be used to solve matrix equations (although I couldn't recommend using the geometric algebra approach for higher dimensions, unless you can do so symbolically).

There's a neat kind of linear algebra (multilinear algebra, technically) that doesn't just deal with vectors (think vectors and planes and volumes), but it's often computationally unweildy. This is a fast library for it that competes with simple matrix code in efficiency.