Heterogeneous ice nucleation (HIN) is strongly related to the dynamics of hydrogen bonds in water at an interface. In this work, we investigate the microscopic kinetics of HIN through molecular dynamics simulations. The dynamics of hydrogen bond network (HBN) at interfaces is studied under the coupled effects of thermal fluctuation and water-surface molecular interactions. It is revealed that the lasting time of the HBN at the interface is critical to HIN. Under comparable thermal and surface effects, which result in a proper lasting time of the HBN, HIN is promoted. However, if the thermal effect or the surface effect dominates over the other, the lasting time of the HBN at the interface would be either too long or too short, leading to the failure of HIN. By varying the water-surface interaction strength, i.e., binding energy, and temperature, a diagram of HIN events is presented, which shows that HIN is only favored in certain temperature and surface energy ranges.