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 responses of these crystals is of primary importance. Here, we combine model Hamiltonians with multi-reference configuration interaction wavefunctions to accurately determine the strength of the spin couplings in the prototypical single-layer magnet CrI$_3$. 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 CrI$_3$ features an out-of-plane easy axis ensuing from a single-ion anisotropy $A = -0.10$ meV, and predict $g$-tensor in-plane components $g_{xx} = g_{yy} = 1.90$ and out-of-plane component $g_{zz} = 1.92$. 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 experimental observations.
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