The Landau damping (LD) mechanism of the localized surface plasmon (LSP) decay is studied for the hybrid nanoplasmonic (metal core/dielectric shell) structures. It is shown that LD in hybrid structures is strongly affected by the permittivity and the electron effective mass in the dielectric shell in accordance with previous observations by Kreibig, and the strength of LD can be enhanced by an order of magnitude for some combinations of permittivity and effective mass. The physical reason for this effect is identified as an electron spillover into the dielectric where the electric field is higher than that in the metal and the presence of quasi-discrete energy levels in the dielectric. The theory indicates that the transition absorption at the metal-dielectric interface is a dominant contribution to LD in such hybrid structures. Thus, by judicious selection of dielectric material and its thickness, one can engineer decay rates and hot carrier production for important applications, such as photodetection and photochemistry.