The Balloon-borne Large Aperture Submillimeter Telescope (BLAST) has recently surveyed ~8.7deg^2^ centered on Great Observatories Origins Deep Survey-South at 250, 350, and 500um. In Dye et al. (2009, Cat. J/ApJ/703/285), we presented the catalog 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 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.