Reservoir computing has emerged as a powerful framework for time-series modeling and forecasting, including the prediction of discontinuous transitions. However, a mechanistic understanding of how reservoir computing reproduces discontinuous dynamical phenomena remains unexplored. This Letter elucidates the functioning of reservoir computing by analyzing its successful reproduction of boundary and attractor-merging crises. By examining the internal dynamics of the trained reservoir map, we reveal how a reservoir that is dynamically distinct from the target system nonetheless undergoes the same crisis mechanism and reproduces the associated scaling exponent with exact statistical correspondence. We establish this across distinct systems, exemplified by the logistic and Gauss maps. The Letter contributes to a broader understanding of the internal dynamics that enable learning algorithms to anticipate critical transitions.