In rare-earth compounds, the interplay between geometric frustration and unusual single-ion anisotropy caused by the crystal electric field often leads to exotic magnetic phases. Here, we investigate the magnetic properties of the rare-earth Shastry-Sutherland compound Er₂Be₂GeO₇ using bulk measurements and neutron scattering down to sub-Kelvin temperatures. A k=(0,0,0) stripe-like antiferromagnetic order with locally tilted Er³⁺ moments is revealed by time-of-flight neutron powder diffraction, accompanied with a λ-shape specific-heat anomaly at T_N=0.84 K. The temperature dependence of the magnetic peak intensity corresponding to the ordered moment of Er³⁺ approximates to the two-dimensional Ising universality class. Thermal- and cold-neutron inelastic scattering experiments demonstrate that the strong Ising anisotropy originates from the local crystal electric field, and reveal gapped magnetic excitations with weak dispersion in the ordered state. Mean-field random-phase approximation with inequivalent nearest-neighbor couplings accounts well for the collective excitations, while classical Monte Carlo simulations yield the fractional 1/4 and 1/2 magnetization plateaus. Our work provides a comprehensive description of the spin correlations in Er₂Be₂GeO₇ with a particular focus on the classical origin of its fractional magnetization plateaus.