How have we determined that it is some underlying grid or space itself that is expanding rather than the objects just moving apart on a static grid of spacetime?
General relativity tells us that the geometry of spacetime is dynamic, being affected by the matter content and its motion. So a fixed, static background spacetime is by its nature some sort of idealized edge case at best.
Well, let's try it. Imagine that in flat, static Minwkoski spacetime with spherical coordinates $(T,R,\theta,\phi)$, there is an expanding spherically symmetric cloud of galaxies expanding from a center at $T = 0$, each with some velocity $v$, such that the galaxies have a negligible effect on the background geometry or each others' velocity. Thus the radial coordinate of each galaxy is $R = vT$.
Parametrizing by the rapidity $\eta$ at the given radial coordinate ($v = \tanh \eta$) and the time measured by the galaxy $t$ as given by special-relativistic time dilation ($t = T/\gamma = T/\cosh\eta$), in those coordinates, the Minwkoski metric becomes
$$\begin{eqnarray*}
\mathrm{d}s^2 &=& -\mathrm{d}T^2 + \mathrm{d}R^2 + R^2\mathrm{d}\Omega^2\\
&=& -\mathrm{d}t^2 + t^2\left(\mathrm{d}\eta^2 + \sinh^2 \eta\,\mathrm{d}\Omega^2\right)\text{,}
\end{eqnarray*}$$
which is a linearly expanding spatially hyperbolic universe where $\eta$ plays the role of the radial coordinate (up to some dimensionful factor, i.e., $r = r_0\eta$).
We sure didn't get far! A spherically symmetric explosion of galaxies in a fixed, static Minkowski spacetime is equivalent to a spatially hyperbolic universe that in which space itself is expanding, with a scale factor $a\propto t$ in terms of cosmological time $t$.
For a homogeneous and isotropic universe, the time-projection of the Einstein field equation is the first Friedmann equation,
$$\frac{\dot{a}^2+k}{a^2} = \frac{8\pi G\rho + \Lambda}{3}\text{,}$$
so in the case of vanishing cosmological constant ($\Lambda = 0$) and negligible energy density ($\rho = 0$), its solution $a(t) = a_0\pm it\sqrt{k}$ is real only if $k<0$, implying an open, spatially hyperbolic universe; conventionally, $k\in\{-1,0,+1\}$, as only the sign of $k$ is important.
Why propose an expanding underlying grid instead of some other mystery force that accelerates objects away faster the farther away they are from us? How would we distinguish between those two scenarios?
If one introduces an extra force and fine-tunes things well enough, you probably can't. However, this is an incredibly silly thing to do: you would need to postulate that (1) gravity simply does not work on the cosmological scale, and that (2) that its effects are exactly mimicked by an extra force that we have absolutely no evidence for. This doesn't achieve anything, not even philosophical pandering, because we would still know that general relativity is correct on smaller scales, so spacetime geometry would still be dynamic on those scales.
