Magnetic anisotropy plays a vital role in stabilizing the long-range magnetic order of two-dimensional ferromagnetic systems. In this work, using the first-principles method, we systematically explored the triaxial magnetic anisotropic properties of a ferromagnetic semiconductor CrSBr monolayer, which is recently exfoliated from its bulk. Further analysis shows that the triaxial magnetic anisotropic properties originate from the coexistence of the magnetic dipole-dipole interaction (shape anisotropy) and the spin-orbit coupling interaction (magnetocrystalline anisotropy). Interestingly, the shape anisotropy, which has been neglected in most previous works, dominates over the magnetocrystalline anisotropy. Besides, the experimental Curie temperature of the CrSBr monolayer is well reproduced using Monte Carlo simulations. What is more, the easy magnetic axes and ferromagnetism in the CrSBr monolayer can be manipulated by strains and are relatively more susceptible to the uniaxial strain in the x direction. Our study not only explains the mechanism of triaxial magnetic anisotropy of the CrSBr monolayer, but also sheds light on how to tune the magnetic anisotropy and Curie temperature in ferromagnetic monolayers.