The search for topological spin excitations in recently discovered two-dimensional (2D) van der Waals (vdW) magnetic materials is important because of their potential applications in dissipationless spintronics. In the 2D vdW ferromagnetic (FM) honeycomb lattice CrI3 (TC=61 K), acoustic and optical spin waves are found to be separated by a gap at the Dirac points. The presence of such a gap is a signature of topological spin excitations if it arises from the next-nearest-neighbor (NNN) Dzyaloshinskii-Moriya (DM) or bond-angle-dependent Kitaev interactions within the Cr honeycomb lattice. Alternatively, the gap is suggested to arise from an electron correlation effect not associated with topological spin excitations. Here, we use inelastic neutron scattering to conclusively demonstrate that the Kitaev interactions and electron correlation effects cannot describe spin waves, Dirac gaps, and their in-plane magnetic field dependence. Our results support the idea that the DM interactions are the microscopic origin of the observed Dirac gap. Moreover, we find that the nearest-neighbor (NN) magnetic exchange interactions along the c axis are antiferromagnetic (AF), and the NNN interactions are FM. Therefore, our results unveil the origin of the observed c-axis AF order in thin layers of CrI3, firmly determine the microscopic spin interactions in bulk CrI3, and provide a new understanding of topology-driven spin excitations in 2D vdW magnets.