The influence of charge trap states in the dielectric boundary material on capacitively coupled radio-frequency (RF) plasma discharge is investigated with theory and particle-in-cell/Monte Carlo collision simulation. It is found that the trap states of the wall material manipulated discharge properties mainly through the varying ion-induced secondary electron emission (SEE) coefficient in response to dynamic surface charges accumulated within the solid boundary. A comprehensive SEE model considering surface charging is established first, which incorporates the valence band electron distribution, electron trap density, and charge trapping through Auger neutralization and de-excitation. Theoretical analysis is carried out to reveal the effects of trap states on sheath solution, stability, plasma density and temperature, particle and power balance, etc. The theoretical work is supported by simulation results, showing the reduction of the mean RF sheath potential as charging-dependent emission coefficient increases. As the gas pressure increases, a shift of the maximum ionization rate from the bulk plasma center to the plasma-sheath interface is observed, which is also influenced by the trap states of the electrode material where the shift happens at a lower pressure with traps considered. In addition, charge traps are proven to be helpful for creating asymmetric plasma discharges with geometrically symmetric structures; such an effect is more pronounced in gamma-mode discharges.