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We report the experimental observation of the spin-wave moiré edge and cavity modes using Brillouin light scattering spectromicroscopy in a nanostructured magnetic moiré lattice consisting of two twisted triangle antidot lattices based on an yttrium iron garnet thin film. Spin-wave moiré edge modes are detected at an optimal twist angle and with a selective excitation frequency. At a given twist angle, the magnetic field acts as an additional degree of freedom for tuning the chiral behavior of the magnon edge modes. Micromagnetic simulations indicate that the edge modes emerge within the original magnonic band gap and at the intersection between a mini flatband and a propagation magnon branch. Our theoretical estimate for the Berry curvature of the magnon-magnon coupling suggests a nontrivial topology for the chiral edge modes and confirms the key role played by the dipolar interaction. Our findings shed light on the topological nature of the magnon edge mode for emergent moiré magnonics.