The nature of the gap observed at the zone border in the spin excitation spectrum of CrI3 quasitwo-dimensional single crystals is still controversial. We perform first-principles calculations based on time-dependent density functional perturbation theory, which indicate that the observed gap results from a combination of spin-orbit and interlayer interaction effects. The former give rise to the anisotropic spin-spin interactions that are responsible for its very existence, while the latter determine both its displacement from the K point of the Brillouin zone, due to the in-plane lattice distortions induced by them, and an enhancement of its magnitude, in agreement with experiments and previous theoretical work based on a lattice model.