The unsteady three-dimensional incompressible Navier-Stokes equations are solved numerically to study the mechanisms that lead to the transition from steady bubble-type vortex breakdown to unsteady columnar vortex flow in a closed cylinder with a rotating lid. Calculations are carried out to simulate the impulsive acceleration of the flow from Reynolds number Re=2100, where the flow is steady with two distinct breakdown bubbles, to Re=3750. The computed results show that the widely used, in previous numerical simulations of this flow, assumption of axisymmetric flow is not valid even for the initial steady flowfield (Re=2100). Three-dimensionally originates inside the centrifugally unstable Stewartson layer in the form of spiral disturbances. During the early stages of the impulsive acceleration, the flow is dominated by the centrifugal instability of the Stewartson layer that casues the initial spiral disturbances to evolve rapidly into spiral Taylor-Gortler vortices. These vortices, which originate just downstream of the rotating wall, spread rapidly towards the stationary wall and begin to interact with the swirling flow along the axis. This interaction is shown to be responsible for the collapse of the breakdown bubbles and the formation of a columnar vortex core subject to moving wave trains. The present results are consistent with recent linear stability findings and are further supported by available experiments.