Unlike the dislocation-based plasticity in crystalline metals, which can be readily explained by their crystal structure and the presence of defects, the nature of the plasticity in amorphous alloys is not completely understood. Experiments have shown that the plasticity in amorphous alloys is strongly dependent on their atomic packing density. This study, based on the combination of experimental and computational techniques, examines the origin of the plasticity in amorphous alloys considering characteristics of the inherent atomic-scale structure as defined by short-range ordered (SRO) clusters. The role of various SRO atomic clusters in creating free volume during shear deformation is discussed. We report that the plasticity exhibited by amorphous alloys is very sensitive to the characteristics of the atomic packing state, which can be described by various SRO atomic structures and quantified by the effective activation energy for crystallization.