(Abridged) This is the second in a series of three papers which present and interpret basic observational data on the 6C* 151-MHz radio sample: a low-frequency selected sample which exploits filtering criteria based on radio properties (steep spectral index and small angular size) to find radio sources at redshift z > 4 within a 0.133sr patch of sky. We present results of a programme of optical spectroscopy which has yielded redshifts in the range 0.5 < z < 4.4 for the 29 sources in the sample, all but six of which are secure. We find that the fil tering criteria used for 6C* are very effective in excluding the low-redshift, low-luminosity radio sources: the median redshift of 6C* is z~1.9 compared to z~1.1 for a complete sample matched in 151-MHz flux density. By combining the emission-line dataset for the 6C* radio sources with those for the 3CRR, 6CE and 7CRS samples we establish that z > 1.75 radio galaxies follow a rough proportionality between Lyalpha- and 151 MHz-luminosity which, like similar correlations seen in samples of lower-redshift radio sources, are indicative of a primary link between the power in the source of the photoionising photons (most likely a hidden quasar nucleus) and the power carried by the radio jets. We argue that radio sources modify their environments and that the range of emission-line properties seen is determined more by the range of source age than by the range in ambient environment. This is in accord with the idea that all high-redshift, high-luminosity radio sources are triggered in similar environments, presumably recently collapsed massive structures.