We determine the importance of extreme, record-sized events on the
non-equilibrium relaxation ("aging") after a sudden quench into the glassy
phase. Here, we directly measure the impact of extreme events on the evolving
system in Monte Carlo simulations of Ising spin glasses and ferromagnets
undergoing quenches into either a low or high-temperature phase. Our results
show that, if we bar the attainment of new record-high energy fluctuations (by
explicitly imposing a "lid" on the fluctuation spectrum), further relaxation in
the low-temperature glassy phase is impeded markedly while in all other phases
such a lid actually accelerates the relaxation process. Such rare record
events, emerging naturally in the sequence of ordinary fluctuations of any
relaxing system, thus prove to be key in activating the aging dynamics, as has
been argued for systems like spin glasses, superconductors, gels, colloids, and
granular piles. This dynamics of records succeeds in explaining the logarithmic
decay of the free energy and the memory effects encoded in the scaling of
two-time correlation functions of those aging systems. These findings are
interpreted through the interplay of fluctuations and generic features of the
hierarchical, complex energy landscape of glassy systems.