Exploiting the Herschel Astrophysical Terahertz Large Area Survey Science Demonstration Phase survey data, we have determined the luminosity functions (LFs) at rest-frame wavelengths of 100 and 250 μm and at several redshifts z >~ 1, for bright submillimeter galaxies with star formation rates (SFRs) >~ 100 M ⊙ yr-1. We find that the evolution of the comoving LF is strong up to z ≈ 2.5, and slows down at higher redshifts. From the LFs and the information on halo masses inferred from clustering analysis, we derived an average relation between SFR and halo mass (and its scatter). We also infer that the timescale of the main episode of dust-enshrouded star formation in massive halos (M H >~ 3 × 1012 M ⊙) amounts to ~7 × 108 yr. Given the SFRs, which are in the range of 102-103 M ⊙ yr-1, this timescale implies final stellar masses of the order of 1011-1012 M ⊙. The corresponding stellar mass function matches the observed mass function of passively evolving galaxies at z >~ 1. The comparison of the statistics for submillimeter and UV-selected galaxies suggests that the dust-free, UV bright phase is >~ 102 times shorter than the submillimeter bright phase, implying that the dust must form soon after the onset of star formation. Using a single reference spectral energy distribution (SED; the one of the z ≈ 2.3 galaxy SMM J2135-0102), our simple physical model is able to reproduce not only the LFs at different redshifts >1 but also the counts at wavelengths ranging from 250 μm to ≈1 mm. Owing to the steepness of the counts and their relatively broad frequency range, this result suggests that the dispersion of submillimeter SEDs of z > 1 galaxies around the reference one is rather small. Herschel is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA.