Spectral geometry is one of the many ways mathematicians think about geometry. The general idea is that if you have some manifold equipped with a metric, you can cook up some canonical differential operators. These operators can be thought of as linear operators, acting on (infinite-dimensional) vector spaces of functions, tensors, spinors, and the like. Each such linear operator will have a set of eigenvalues. Spectral geometry is concerned with relationships between these eigenvalues and the geometry of the manifold you started with
The most obvious linear operator to associate to a metric is the Laplacian, which is a linear operator on the space of functions on the manifold. In the early/middle part of the 20th century, mathematicians started wondering "If you know the set of eigenvalues of the Laplacian, can you reconstruct the manifold?", or as Mark Kac famously put it: "Can one hear the shape of a drum?"
The answer is no; the set of eigenvalues alone doesn't let you reconstruct the manifold and its metric. The map which sends a manifold with metric to the set of eigenvalues of the Laplacian is not invertible.
But it's such a pretty idea that people haven't given up on it. Alain Connes, for example, figured out that the answer to a slightly different question is "yes". If you have a commutative spectral triple (basically, the Dirac operator on a compact spin manifold), you can reconstruct the metric from this data.
The physicists interviewed in the linked article are trying a slightly different variation on the spectral geometry problem. They're considering systematic finite-dimensional approximations to the "derivative" of the map $F$ which sends a manifold to its set of eigenvalues of its Laplacian acting on tensors of low-degree, and trying to show that these approximations are invertible. This should let them show that this map $F$ is invertible in small regions of the space of manifolds.
It's a nice idea, and looks like some fun experimental mathematics. Trying to write down a theory of gravity in explicitly gauge invariant terms is also a good idea. But I'd be quite surprised if this line of thinking bears any fruit. It seems more likely to me that we need some essentially new physical ideas than a clever way of rewriting what we already have.