The cosmological principle assumes that the universe is statistically homogeneous and isotropic on sufficiently large spatial scales. It applies to a coarse-grained description, not to each galaxy or local void. Combined with the Copernican principle, cosmological isotropy around us supports a homogeneous description rather than a specially centred model. It motivates an FRW metric, whose expansion is governed by a few time-dependent quantities. Observations sample only a finite past light cone, so they support this hypothesis without proving global spatial homogeneity beyond the observed region.
Cosmological isotropy means that coarse-grained spatial observations or their statistics are unchanged by rotations about an observer. In an FRW metric it holds about every comoving location. cosmological isotropy about a single location is weaker than spatial homogeneity: a radially inhomogeneous model centred on that location can also have an isotropic sky.
Our observer location is not specially privileged. In cosmology this assumption allows cosmological isotropy observed around us to be interpreted statistically as cosmological isotropy around other typical observers. It is an assumption about typicality, not a deduction from one observer's sky or from the Einstein field equations alone.
