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Chirality-induced spin selectivity (CISS) generates giant spin polarization in transport through chiral molecules, paving the way for novel spintronic devices and enantiomer separation. Unlike conventional transport, CISS magnetoresistance (MR) violates Onsager's reciprocal relation, exhibiting significant resistance changes when reversing electrode magnetization at zero bias. However, its underlying mechanism remains unresolved. In this work, we propose that CISS MR originates from charge trapping that modifies the electron tunneling barrier and circumvents Onsager's relation, distinct from previous spin polarization-based models. Charge trapping is governed by the non-Hermitian skin effect, where dissipation leads to exponential wavefunction localization at the ferromagnet-chiral molecule interface. Reversing magnetization or chirality alters the localization direction, changing the occupation of impurity/defect states in the molecule (i.e., charge trapping) -- a phenomenon we term magnetochiral charge pumping. Our theory explains why CISS MR can far exceed the ferromagnet spin polarization and why chiral molecules violate the reciprocal relation but chiral metals do not. Furthermore, it predicts exotic phenomena beyond the conventional CISS framework, including asymmetric MR induced by magnetic fields alone (without ferromagnetic electrodes), as confirmed by recent experiments. This work offers a deeper understanding of CISS and opens avenues for controlling electrostatic interactions in chemical and biological systems through the magnetochiral charge pumping.