Ram pressure stripping (RPS) is a process that removes the interstellar medium (ISM) quickly, playing a vital role in galaxy evolution. Previous RPS studies have treated the ISM as single-phase or lack the resolution and physical processes to properly capture the full multiphase ISM. To improve this simplification, we introduce an inflowing, hot intracluster medium (ICM) into a self-consistently modeled ISM in a local patch of star-forming galactic disks using the TIGRESS framework. Our simulations reveal that the workings of RPS are not only direct acceleration of the ISM by ICM ram pressure but also mixing-driven momentum transfer involving significant phase transition and radiative cooling. The hot ICM passes through the low-density channels of the porous, multiphase ISM, shreds the cool ISM, and creates mixing layers. The ICM momentum is transferred through the mixing layers while populating the intermediate temperature gas and radiating thermal energy away. The mixed gas extends beyond galactic disks and forms stripped tails that cool back unless the ICM fluxes are large enough to prevent cooling until they escape the simulation domain. The mixing-driven momentum transfer predicts that the more ICM mixes in, the faster the ISM moves, resulting in the anti-correlation of outflow velocity and gas metallicity of the stripped ISM. The compression of the ISM disks due to the ICM ram pressure enhances star formation rates up to 50% compared to the model without ICM. With the ICM ram pressure higher than the disk anchoring pressure, star formation is quenched within ~100 Myr.