So we observe that the Universe is expanding. We observe galaxies moving away from us and the speed/red-shift is greater for galaxies that are further away.
How do we know that this observation is the space metric expanding as opposed to objects moving relative to one another in a static metric?
We observe that galaxies appear to move away from us, in an isotropic fashion, at a rate that is proportional to their distance from us (Hubble's law).
Whilst one could argue that we are at (or near) the centre of a very uniform expansion, one would have to explain why Hubble's law should exist and why the universe appears isotropic to us, but wouldn't from a different position in the universe. The simplest explanation is that General Relativity applies (as we observe in a number of other cases) and we live in an expanding universe - this then means we do not need to occupy some privileged position in the universe (an erroneous assumption that has proved wrong every other time it has been made).
In such a universe, the redshift of distant galaxies is not caused by relative motion, but by the expansion of space. At high redshifts, these phenomena become distinct in that the relationship between "velocity" and redshift is different, for instance allowing "faster than light" (apparent) speeds.
So basically at present, expansion fits the facts (far) better and more simply than any of the alternatives.
A further piece of indirect evidence comes from a careful analysis of the physical conditions of gas at high redshifts, illuminated by background quasars and subtle alterations to the cosmic microwave background (CMB) spectrum, caused by the Sunyaev-Zel'dovich effect, towards galaxy clusters at low redshifts. Both of these methods give the temperature of the CMB at those locations.
In the expanding universe model, the temperature should increase as $1+z$, where $z$ is the redshift. If one instead has a non-expanding universe, and explain the CMB as due to some expanding shell of material, then the average temperature wouldn't change for distant galaxies unless the shell gas has been uniformly cooling by an amount that just happens to agree with redshift of that galaxy.
Avgoustidis et al. (2015) review the evidence for the temperature evolution of the CMB and conclude that it agrees with an adiabatic expansion to better than 1%.
Direct evidence for the expansion is on the horizon though. In an expanding universe, the speed at which galaxies move away from us can change slowly with time (and with distance) by of order 10 cm/s per year, despite their being no force on them. This is known as the redshift drift. There are plans to measure this tiny effect with the European Extremely Large Telescope and the Square Kilometre Array over the course of a decade.
The cosmological model is a mathematical model built on General Relativity. The metric of general relativity is four dimensional.
The observation of "everything moving away from everything else" cannot be fitted in three dimensional space logically. It looks as if there is an explosion everywhere, in every (x,y,z). Why need an explosion? To explain how the energy came to be. So it cannot be fitted in three dimensional space gravity with a reasonable model.
A calculable two dimensional mathematical model, of everything moving away from everything else can be found in the balloon analogy. As a balloon expands, everypoint on its surface moves away from all other points. Unfortunately, we observe the galaxy expansion in three dimensional space and we need a fourth dimension to have the mathematical analogue. Which brings in the model of General relativity, which carries the concept of space expansion as the simplest mathematical formulation to explain why everything moves away from everything else. Mind you, GR fits all observations and experimental data up to now.
As I'd mentioned in a comment on the question itself, Davis (of the Lineweaver and Davis pair who have plotted various observational horizons in much-used diagrams) has made clear that spatial expansion is not a "force or drag" carrying other material and energetic objects along with it. Nevertheless, space expands locally as a result of changes in the number or trajectories of the massive and/or or energetic objects it contains, which are, themselves, interchangeable with each other according to the mass / energy equivalence of General Relativity: In effect, changes of the spatial contents, from mass to energy, can result in its expansion. (Although an infinite universe would be inherently unverifiable, it might exist, with a locally-apparent "expansion" actually comprising a "subdivision" of its space.)
Cosmological models that explain a locally-apparent expansion of space by using 1915's General Relativity alone tend to rely on "dark energy" or "repulsive gravity" for driving the astronomical objects apart and thereby explaining the redshift. However, the model which takes the most advantage of Einsteinian relativity is Nikodem Poplawski's "cosmology with torsion" (described in numerous papers, written between 2010 and 2021, whose preprints can be found by his name on Cornell University's "Arxiv" website): It uses Einstein-Cartan Theory, (developed by Einstein and the French mathematician Elie Cartan in 1929, a few years after the discovery of particulate spin), which assigns a tiny spatial extent to fermions.
His model applies these considerations in the context of the gravitational collapse of large stars into black holes, after their exhaustion of their nuclear fuel has left them with insufficient radiation pressure to prevent it. (As most stars are in binary pairs, there's astronomical evidence of such collapses having occurred at least 90 times within our observable region, as the surviving star of each such pair continues to follow the elliptical orbit formerly shared with its partner.)
In Poplawski's model, the trajectories of fermions newly-materialized by the gravitational field (thru their separation from partners in virtual particle/ antiparticle pairs by the outward-propagating event horizon of the collapsing star) are reversed and greatly accelerated thru interaction with the vastly larger stellar fermions, and form a new "local universe" whose shape Poplawski has analogized to the skin of a basketball.
If a universe old enough would be infinite (or even just "extremely large") in spatial extent, and about as homogeneous and isotropic everywhere as it appears to be in the portion of it observable from earth, life in its central region would be less probable without such causal separations as are permitted by the "Black Hole Genesis" suggested by Lee Smolin and implemented by Poplawski, as the light eventually arriving from its peripheral regions would raise planetary surface temperatures beyond the boiling point, which leaves the formation of stable structures much less probable. A good summary of BHG is freely visible online, in the video "Before the Big Bang 10 : Black Hole Genesis".
The addition of varying amounts of waste or other mass, in some cases quite small, to large stars otherwise apt to collapse into neutron stars, may have conceivably been undertaken by advanced civilizations so as to cause them to collapse into black holes instead, in the expectation that one or another of the LU's germinating inside them would resemble such a civilization's own home, in a form of intermittent & approximate self-perpetuation not possible thru inflation based on scalar fields: This possibility may have left the ECT-based model more probable.
"Black Hole Genesis" developed as a factor within the larger framework of Smolin's "Cosmological Natural Selection": Consequently, it occurs to me that, just as a sufficiently large population of any biological species will tend to provide a breeding ground for the evolution of parasites apt to reduce it, the "artificial cosmological selection" sketched in my previous paragraph might tend to reduce the proportion of neutron stars vis-a-vis BH's, but increase their size during whatever additions of waste or other mass might have occurred within cosmologically recent time, given the possibility that the hypothetical civilizations involved might reasonably be expected to have a broader spectrum of data as to exactly what conditions and surroundings might influence the break point between the two types of astrophysical objects under consideration.
A different past- and future-eternal cosmological model (which, like Poplawski's, dates from 2010) was written by the mathematical physicist Roger Penrose, a frequent collaborator of Stephen Hawking, and a winner of 2020's Nobel Prize for physics. In his "cyclic conformal cosmology", which depends on gravitational entropy functioning inversely to thermodynamic entropy, the thermal equilibrium of each of an endless sequence of "aeons" becomes the "big bang" of the next after the gravitational decoherence of all of the previous aeon's massive particles into radiation. (Conformal geometry, preserving angles during huge increases in scale, permits the diagrammatic representation of this interesting model, whose contents, when consisting only of radiation, would lack any physical time, because of the fact that all known or imaginable clocks would require components having mass.)
There has been consistent evidence for the Penrose model in the observation of "Hawking points" (anomalous spots of significantly raised temperature in the CMB sky that result from the release of Hawking radiation by the "evaporation" of black holes), as discussed in the March 2020 paper at https://arxiv.org/abs/1808.01740.
