The air bubble entrainment and self-aeration phenomena in free-surface water flows reveal a rich interplay of fundamental science and engineering, and the size distribution of the entrained bubbles enhances the air-water gas flux, improves the gas transfer, and influences the cavitation erosion protection in high-speed flows. In the present study, we investigate the bubble-formation mechanism of free-surface air entrainment and the related microscopic bubble scale in the laboratory. This paper provides a quantitative description of bubble entrainment. The entrapment deformation of the local free surface over a period follows a power-law scaling and entrains a bubble when the entrapped surface becomes enclosed in the unstable movement period. Both the size scale and shape character of the entrapped free surface determine the size and skewness of the distribution of the air bubble. The entrapment deformation process confirms that the instability behaviour of the local air-water interface results in the onset of bubble entrainment. Further research is necessary to elucidate the instability criterion dominated by the interface instability and promote a new understanding of multiphase flow generation and development.