We describe the Carnegie-Spitzer-IMACS (CSI) Survey, a wide-field, near-IR selected spectrophotometric redshift survey with the Inamori Magellan Areal Camera and Spectrograph (IMACS) on Magellan-Baade. By defining a flux-limited sample of galaxies in Spitzer Infrared Array Camera 3.6 mu m imaging of SWIRE fields, the CSI Survey efficiently traces the stellar mass of average galaxies to z similar to 1.5. This first paper provides an overview of the survey selection, observations, processing of the photometry and spectrophotometry. We also describe the processing of the data: new methods of fitting synthetic templates of spectral energy distributions are used to derive redshifts, stellar masses, emission line luminosities, and coarse information on recent star formation. Our unique methodology for analyzing low-dispersion spectra taken with multilayer prisms in IMACS, combined with panchromatic photometry from the ultraviolet to the IR, has yielded high-quality redshifts for 43,347 galaxies in our first 5.3 deg(2) of the SWIREXMM-LSS field. We use three different approaches to estimate our redshift errors and find robust agreement. Over the full range of 3.6 mu m fluxes of our selection, we find typical redshift uncertainties of sigma(z)/(1 + z) less than or similar to 0.015. In comparisons with previously published spectroscopic redshifts we find scatters of sigma z/(1 + z) = 0.011 for galaxies at 0.7 <= z <= 0.9, and sigma z/(1 + z) = 0.014 for galaxies at 0.9 <= z <= 1.2. For galaxies brighter and fainter than i = 23 mag, we find sigma z/(1 + z) = 0.008 and sigma z/(1 + z) = 0.022, respectively. Notably, our low-dispersion spectroscopy and analysis yields comparable redshift uncertainties and success rates for both red and blue galaxies, largely eliminating color-based systematics that can seriously bias observed dependencies of galaxy evolution on environment.