The Probe Of Extreme Multi-Messenger Astrophysics (POEMMA) is a potential NASA Astrophysics Probe-class mission designed to observe ultra-high energy cosmic rays (UHECRs) and cosmic neutrinos from space. POEMMA will monitor colossal volumes of the Earth's atmosphere to detect extensive air showers (EASs) produced by extremely energetic cosmic messengers: UHECRs above 20 EeV over the full sky and cosmic neutrinos above 20 PeV. We focus most of this study on the impact of POEMMA for UHECR science by simulating the detector response and mission performance for EAS from UHECRs. We show that POEMMA will provide a significant increase in the statistics of observed UHECRs at the highest energies over the entire sky. POEMMA will be the first UHECR fluorescence detector deployed in space that will provide high-quality stereoscopic observations of the longitudinal development of air showers. Therefore, it will be able to provide event-by-event estimates of the calorimetric energy and nuclear mass of UHECRs. The particle physics in the interactions limits the interpretation of the shower maximum on an event by event basis. In contrast, the calorimetric energy measurement is significantly less sensitive to the different possible final states in the early interactions. We study the prospects to discover the origin and nature of UHECRs using expectations for measurements of the energy spectrum, the distribution of arrival direction, and the atmospheric column depth at which the EAS longitudinal development reaches maximum. We also explore supplementary science capabilities of POEMMA through its sensitivity to particle interactions at extreme energies and its ability to detect ultra-high energy neutrinos and photons produced by top-down models including cosmic strings and super-heavy dark matter particle decay in the halo of the Milky Way.