Our cosmology course instructor said that
At a given cosmic time $t=\tau$, the comoving observers define a spacelike hypersurface $\Sigma$ in which the universe looks homogenous and isotropic. Another observer, who is moving with a uniform velocity w.r.t the class of comoving observers will find the universe to be anisotropic. An intergalactic spaceship moving at a high speed would see the galactic distribution quite differently.
I have several questions based on his statement.
What does he mean by "the comoving observers define a spacelike hypersurface $\Sigma$."?
Is it that the homogeneity and isotropy of the Universe true only for the comoving observers? If yes, why?