The effect of local structures on structural evolution during the crystallization of undercooled ZrCu metallic glass-forming liquid was studied via molecular dynamics simulations. It is found that body-centered-cubic (bcc)-like clusters play a key role in structural evolution during crystallization. In contrast to previous speculations, the number of bcc-like crystal nuclei does not change much before the onset of crystallization. Instead, the development of a bcc-like critical nucleus during annealing leads to a strong spatial correlation with other nuclei in its surroundings, forming a crystalline structure template. It is also found that the size distribution of bcc-like nuclei follows a power-law form with an exponential cutoff in the early stage of annealing, but changes to a pure power-law behavior just before the onset of crystallization. This implies that the crystalline structure template has fractal feature and the undercooled liquids evolve to a self-organized critical state before the onset of crystallization, which might trigger the subsequent rapid crystallization. According to the graph theory analysis, it is also found that the observed large scatter of the onset time of crystallization in different liquid samples results from the connectivity of the bcc-like clusters.