The Balloon-borne Large Aperture Submillimeter Telescope (BLAST) has recently surveyed ~8.7 deg^2 centered on GOODS-South at 250, 350, and 500 microns. In Dye et al. (2009) we presented the catalogue of sources detected at 5-sigma in at least one band in this field and the probable counterparts to these sources in other wavebands. In this paper, we present the results of a redshift survey in which we succeeded in measuring redshifts for 82 of these counterparts. The spectra show that the BLAST counterparts are mostly star-forming galaxies but not extreme ones when compared to those found in the Sloan Digital Sky Survey. Roughly one quarter of the BLAST counterparts contain an active nucleus. We have used the spectroscopic redshifts to carry out a test of the ability of photometric redshift methods to estimate the redshifts of dusty galaxies, showing that the standard methods work well even when a galaxy contains a large amount of dust. We have also investigated the cases where there are two possible counterparts to the BLAST source, finding that in at least half of these there is evidence that the two galaxies are physically associated, either because they are interacting or because they are in the same large-scale structure. Finally, we have made the first direct measurements of the luminosity function in the three BLAST bands. We find strong evolution out to z=1, in the sense that there is a large increase in the space-density of the most luminous galaxies. We have also investigated the evolution of the dust-mass function, finding similar strong evolution in the space-density of the galaxies with the largest dust masses, showing that the luminosity evolution seen in many wavebands is associated with an increase in the reservoir of interstellar matter in galaxies. ; Comment: Accepted for publication in the Astrophysical Journal. Maps and associated results are available at http://blastexperiment.info/