We present a statistical analysis of the first 300 stars observed by the Gemini Planet Imager Exoplanet Survey. This subsample includes six detected planets and three brown dwarfs; from these detections and our contrast curves we infer the underlying distributions of substellar companions with respect to their mass, semimajor axis, and host stellar mass. We uncover a strong correlation between planet occurrence rate and host star mass, with stars M_*_>1.5 M_{sun}_ more likely to host planets with masses between 2 and 13 M_Jup_ and semimajor axes of 3-100 au at 99.92% confidence. We fit a double power-law model in planet mass (m) and semimajor axis (a) for planet populations around high-mass stars (M_*_>1.5 M_{sun}_) of the form d^2^N/(dm da){prop.to}m^{alpha}^{alpha}^{beta}^, finding {alpha}=-2.4+/-0.8 and {beta}=-2.0+/-0.5, and an integrated occurrence rate of 9_-4_^+5^% between 5-13 M_Jup_ and 10-100 au. A significantly lower occurrence rate is obtained for brown dwarfs around all stars, with 0.8_-0.5_^+0.8^% of stars hosting a brown dwarf companion between 13-80 M_Jup_ and 10-100 au. Brown dwarfs also appear to be distributed differently in mass and semimajor axis compared to giant planets; whereas giant planets follow a bottom-heavy mass distribution and favor smaller semimajor axes, brown dwarfs exhibit just the opposite behaviors. Comparing to studies of short-period giant planets from the radial velocity method, our results are consistent with a peak in occurrence of giant planets between ~1 and 10 au. We discuss how these trends, including the preference of giant planets for high-mass host stars, point to formation of giant planets by core/pebble accretion, and formation of brown dwarfs by gravitational instability.