The inside of the light cone is the observable universe: as time goes by more and more of the universe becomes observable, and the volume of the past lightcone increases. The limit to our observable universe at any time, defined by the finite extent of the past light cone, defined our particle horizon (Notes section 2.15).
Out worldline tracks up the central time axis at the apex of the lightcone. Worldlines of distant galaxies are represented by the curves moving upwards whose proper distance relative to us is increasing as the universe expands. We receive light signals from these galaxies sent along the lightcone; at any time we see the nearer galaxy as it was some time in the past, and the further galaxy as it was even more distant in the past.