The successful development of all-solid-state batteries will provide solutions for many problems facing current Li-ion batteries, such as high flammability, limited energy density, poor cyclability and low cation transference number. In this quest, the development of high-performance solid-state electrolytes is critical. Composite polymer electrolytes (CPE), comprising ion-conducting (active) inorganic fillers and polymer matrices, have emerged as a promising strategy to yield better conductivity, interfacial stability, and mechanical strength than their single-phase counterparts. Recent experiments indicate that active garnet fillers may enhance the ionic conductivity of CPEs by inducing anion trapping onto their surface. Moreover, substitutions that modify the lithium molar content within the filler were shown to impact this enhancement. However, the molecular underpinning behind this phenomenon is poorly understood, hindering the development of strategies to exploit it optimally. In this study, we use an enhanced hybrid Monte Carlo technique in combination with extensive molecular dynamics simulations to bridge this gap. By focusing on the archetypal CPE formed by Ga-doped Li7-3xGaxLa3Zr2O12 (Gax-LLZO) embedded within a poly(ethylene oxide) (PEO) and lithium bis(trifluoromethane sulfonyl) imide (LiTFSI) polymer matrix, we describe how the dynamic electrostatic trapping of anions leads to overall conductivity enhancement by increasing the lithium transference number and tracer diffusivity in the polymer phase. The extent of this enhancement can be fine-tuned by modulating the Li molar content of LLZO through the doping of Ga. We predict an optimal Li molar content of 5.95, which is lower than the optimal 6.50 reported in the literature for single LLZO.