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We theoretically investigate the light-induced excitation of two-dimensional surface polaritons (2DSPs) at the edge of atomically thin crystals. Through our analytic model based on the coupled mode theory, we achieve a precise and quantitative description of light interaction with the crystal's edge. Our model unveils three notable properties associated with the edge-excitation of 2DSPs. Firstly, under strong confinement conditions (i.e., polariton momentum significantly surpassing free-space photon momentum), the amplitude of edge-excited 2DSPs exhibits a cosine dependency on the incident angle. Secondly, the amplitude reaches a saturation point at a specific value as polariton momentum increases, depending on the incident angle. Lastly, we demonstrate that the edge-excitation of 2DSPs entails the generation of induced unbounded waves near the edge, leading to peak oscillations due to the interference between the 2DSPs and the unbounded waves.