We report a thorough theoretical investigation of magnon-assisted photon transitions in magnetic garnet micron-sized spheres, which operate as optomagnonic resonators. In this case, matching the intraband splitting of optical Mie modes, induced by particle magnetization, to the eigenfrequency of the uniform-precession spin wave, high-efficiency triply resonant optical transitions between these modes, through respective emission or absorption of a cavity magnon, are enabled. By means of rigorous full electrodynamic computations, supported by corresponding approximate analytical calculations, we provide compelling evidence of greatly increased optomagnonic interaction, compared to that in similar processes between whispering gallery modes of corresponding submillimeter spheres, due to the reduced magnon mode volume. We explain the underlying mechanisms to a degree that goes beyond existing interpretation, invoking group theory to derive general selection rules and highlighting the role of the photon spin as the key property for maximizing the respective coupling strength.