Two new entangled Cu(II)-based metal–organic frameworks (MOFs) have been synthesized, namely [Cu(BDC)(BPDB)0.5]n (PNU-25) and [Cu(NH2-BDC)(BPDB)0.5]n (PNU-25-NH2), using a H2O-MeOH solvent mixture. Both the PNU-25 and PNU-25-NH2 MOF materials were characterized by various analytical techniques and their catalytic potential of CO2 fixation into cyclic carbonates at an atmospheric pressure, a low reaction temperature, and in the neat conditions were demonstrated. The amine-functionalized PNU-25-NH2 exhibited a significant high conversion of epichlorohydrin (ECH) at the 1 bar of CO2 pressure, at 55 °C, and a moderate catalyst amount (1 mol%), with over 99% selectivity toward the corresponding cyclic carbonate of ECH. The superior catalytic activity of PNU-25-NH2 may be attributed to its high amount of acidic-basic sites and large BET surface area in comparison with the PNU-25. The PNU-25-NH2 catalyst could be reused up to four cycles without compromising its structural integrity and the ECH conversion. The reaction mechanism of CO2 and ECH cycloaddition reaction mediated by the PNU-25-NH2 was investigated in detail based on the experimental inferences and periodic calculations of density functional theory (DFT). The energy barrier of the rate-determining step of the PNU-25-NH2/TBAB-catalyzed reaction was significantly lower than those of the rate-determining steps of un-catalyzed and TBAB-catalyzed reactions.