According to https://en.wikipedia.org/wiki/World_manifold, gravity can be understood as follows:
In accordance with the geometric Equivalence Principle, a world manifold possesses a Lorentzian structure, i.e., a structure group of a frame bundle $FX$ must be reduced to a Lorentz group $SO(1,3)$. The corresponding global section of the quotient bundle $FX/SO(1,3)$ is a pseudo-Riemannian metric $g$ of signature $(+,−−−)$ on $X$. It is treated as a gravitational field in General Relativity and as a classical Higgs field in gauge gravitation theory.
A Lorentzian structure need not exist. Therefore, a world manifold is assumed to satisfy a certain topological condition. It is either a noncompact topological space or a compact space with a zero Euler characteristic. Usually, one also requires that a world manifold admits a spinor structure in order to describe Dirac fermion fields in gravitation theory. There is the additional topological obstruction to the existence of this structure. In particular, a noncompact world manifold must be parallelizable.
To me this approach is easy to understand because it makes gravity feel almost inevitable. However, I have not been able to find any introductory material on this approach.
Specifically, I am interested in understanding this specific sentence:
It is treated as a gravitational field in General Relativity
What are the next steps to get, say the Hilbert-Einstein Lagrangian or the Einstein field equations from the above?