We present the discovery of a new Jovian-sized planet, TOI-3757 b, the lowest-density transiting planet known to orbit an M dwarf (M0V). This planet was discovered around a solar-metallicity M dwarf, using Transiting Exoplanet Survey Satellite photometry and confirmed with precise radial velocities from the Habitable-zone Planet Finder (HPF) and NEID. With a planetary radius of 12.0 R_⊕ and mass of 85.3 M_⊕, not only does this object add to the small sample of gas giants (∼10) around M dwarfs, but also its low density ( g cm⁻³) provides an opportunity to test theories of planet formation. We present two hypotheses to explain its low density; first, we posit that the low metallicity of its stellar host (∼0.3 dex lower than the median metallicity of M dwarfs hosting gas giants) could have played a role in the delayed formation of a solid core massive enough to initiate runaway accretion. Second, using the eccentricity estimate of 0.14 ± 0.06, we determine it is also plausible for tidal heating to at least partially be responsible for inflating the radius of TOI-3757b b. The low density and large scale height of TOI-3757 b makes it an excellent target for transmission spectroscopy studies of atmospheric escape and composition (transmission spectroscopy measurement of ∼ 190). We use HPF to perform transmission spectroscopy of TOI-3757 b using the helium 10830 Å line. Doing this, we place an upper limit of 6.9% (with 90% confidence) on the maximum depth of the absorption from the metastable transition of He at ∼10830 Å, which can help constraint the atmospheric mass-loss rate in this energy-limited regime.