Elucidating the microscopic origin of nematic order in iron-based superconducting materials is important because the interactions that drive nematic order may also mediate the Cooper pairing. Nematic order breaks fourfold rotational symmetry in the iron plane, which is believed to be driven by either orbital or spin degrees of freedom. However, as the nematic phase often develops at a temperature just above or coincides with a stripe magnetic phase transition, experimentally determining the dominant driving force of nematic order is difficult. Here, we use neutron scattering to study structurally the simplest iron-based superconductor FeSe, which displays a nematic (orthorhombic) phase transition at $T_s=90$ K, but does not order antiferromagnetically. Our data reveal substantial stripe spin fluctuations, which are coupled with orthorhombicity and are enhanced abruptly on cooling to below $T_s$. Moreover, a sharp spin resonance develops in the superconducting state, whose energy (~4 meV) is consistent with an electron boson coupling mode revealed by scanning tunneling spectroscopy, thereby suggesting a spin fluctuation-mediated sign-changing pairing symmetry. By normalizing the dynamic susceptibility into absolute units, we show that the magnetic spectral weight in FeSe is comparable to that of the iron arsenides. Our findings support recent theoretical proposals that both nematicity and superconductivity are driven by spin fluctuations.