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We develop an approach to understanding long chaotic transients in networks of excitatory pulse-coupled oscillators. Our idea is to identify a class of attractors, sequentially active firing (SAF) attractors, in terms of the temporal event structure of firing and receipt of pulses. Then all attractors can be classified into two groups: SAF attractors and non-SAF attractors. We establish that long transients typically arise in the transitional region of the parameter space where the SAF attractors are collectively destabilized. Bifurcation behavior of the SAF attractors is analyzed to provide a detailed understanding of the long irregular transients. Although demonstrated using pulse-coupled oscillator networks, our general methodology may be useful in understanding the origin of transient chaos in other types of networked systems, an extremely challenging problem in nonlinear dynamics and complex systems.