The Cosmological Principle doesn't actually imply that the universe is expanding, instead, what this principle implies that the spatial part of the space-time manifold is maximally symmetric. What this actually means that the spatial manifold has a constant curvature everywhere given by Ricci Scalar, $R=-6\kappa$ using this principle along with defining the cosmological standard time $t$ as the time co-ordinate for co-moving cosmological fluids, we can construct metric given by:
$$d s^{2}=- d t^{2}+R^{2}(t)\left[\frac{d r^{2}}{1-\kappa r^{2}}+r^{2}\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right)\right]$$
where $R(t)$ is the scaling factor. Now, whether the universe is contracting/expanding depends on this scaling factor. In order to determine $R(t)$, we have to solve Einstein's field equation which is given by the Friedmann Equation for this metric:
$$\frac{\dot{R}}{R}=-\frac{4 \pi}{3} G (\rho+3p)+\frac{1}{3} \Lambda c^{2}$$
and
$$\left(\frac{\dot{R}}{R}\right)^{2}-\frac{8}{3} \pi G \rho=-\frac{\kappa c^{2}}{R^{2}}$$
$\rho$ being energy density and $p$ being the pressure for cosmological fluids.
So, in short, it is actually the behavior of cosmological fluids that determines the expansion/contraction of the universe. Also, in the above answer, it might be possible that the galaxies have peculiar velocities due to which can't really say "if the galaxies appear to be rushing away from an observer in the Milky Way galaxy, the same would also appear to be to an observer from the Andromeda galaxy".
