Since thermal fluctuations become more important as dimensions shrink, it is expected that low-dimensional magnets are more sensitive to atomic displacement and phonons than bulk systems are. Here we present a fully relativistic first-principles study on the spin-lattice coupling, i.e., how the magnetic interactions depend on atomic displacement, of the prototypical two-dimensional ferromagnet CrI3. We extract an effective measure of the spin-lattice coupling in CrI3, which is up to ten times larger than what is found for bcc Fe. The magnetic exchange interactions, including Heisenberg and relativistic Dzyaloshinskii-Moriya interactions, are sensitive both to the in-plane motion of Cr atoms and out-of-plane motion of ligand atoms. We find that significant magnetic pair interactions change sign from ferromagnetic (FM) to antiferromagnetic (AFM) for atomic displacements larger than 0.16 (0.18) angstrom for Cr (I) atoms. We explain the observed strong spin-lattice coupling by analyzing the orbital decomposition of isotropic exchange interactions, involving different crystal-field-split Cr-3d orbitals. The competition between the AFM t(2g)-t(2g) and FM t(2g)-e(g) contributions depends on the bond angle formed by Cr and I atoms as well as Cr-Cr distance. In particular, if a Cr atom is displaced, the FM-AFM sign changes when the I-Cr-I bond angle approaches 90 degrees. The obtained spin-lattice coupling constants, along with the microscopic orbital analysis, can act as a guiding principle for further studies of the thermodynamic properties and combined magnon-phonon excitations in two-dimensional magnets.