  Hybrid percolation transitions (HPTs) induced by cascading processes have
been observed in diverse complex systems such as $k$-core percolation,
breakdown on interdependent networks and cooperative epidemic spreading models.
Much effort has been devoted to describe the properties of HPTs of individual
systems. Yet the fundamental question about the possible universal mechanism
underlying those HPTs has not been investigated at a microscopic level. Here,
we find that the discontinuity in the order parameter in such HPTs results from
two steps: a durable critical branching (CB) and an explosive, supercritical
(SC) process. In a random network of $N$ nodes at the transition the CB process
persists for $O(N^{1/3})$ time and the remaining nodes become vulnerable. Those
vulnerable nodes are activated then in the short SC process. This crossover
mechanism and scaling behavior are universal for different HPT systems.
