The marked interplay between the crystalline, electronic, and magnetic structure of atomically thin magnets has been regarded as the key feature for designing next-generation magneto-optoelectronic devices. In this respect, a detailed understanding of the microscopic interactions underlying the magnetic response of these crystals is of primary importance. Here, we combine model Hamiltonians with multireference configuration interaction wavefunctions to accurately determine the strength of the spin couplings in the prototypical single-layer magnet CrI3. Our calculations identify the (ferromagnetic) Heisenberg exchange interaction J = -1.44 meV as the dominant term, being the inter-site magnetic anisotropies substantially weaker. We also find that single-layer CrI3 features an out-of-plane easy axis ensuing from a single-ion anisotropy A = -0.10 meV, and predict g-tensor in-plane components gxx=gyy=1.90g(zz) = 1.92. In addition, we assess the performance of a dozen widely used density functionals against our accurate correlated wavefunctions calculations and available experimental data, thereby establishing reference results for future first-principles investigations. Overall, our findings offer a firm theoretical ground to recent experimental observations.