The formation of membraneless organelles in cells commonly occurs via liquid-liquid phase separation (LLPS), and is in many cases driven by multivalent interactions between intrinsically disordered proteins (IDPs). Molecular simulations can reveal the specific amino acid interactions driving LLPS, which is hard to obtain from experiment. Coarse-grained simulations have been used to directly observe the sequence determinants of phase separation but have limited spatial resolution, while all-atom simulations have yet to be applied to LLPS due to the challenges of large system sizes and long time scales relevant to phase separation. We present a novel multiscale computational framework by obtaining initial molecular configurations of a condensed protein-rich phase from equilibrium coarse-grained simulations, and back mapping to an all-atom representation. Using the specialized Anton 2 supercomputer, we resolve microscopic structural and dynamical details of protein condensates through microsecond-scale all-atom explicit-solvent simulations. We have studied two IDPs which phase separate in vitro: the low complexity domain of FUS and the N-terminal disordered domain of LAF-1. Using this approach, we explain the partitioning of ions between phases with low and high protein density, demonstrate that the proteins are remarkably dynamic within the condensed phase, identify the key residue-residue interaction modes stabilizing the dense phase, all while showing good agreement with experimental observations. Our approach is generally applicable to all-atom studies of other single and multi-component systems of proteins and nucleic acids involved in the formation of membraneless organelles.