Recombination via sub-gap trap states is considered a limiting factor in the development of organometal halide perovskite solar cells. Here, we demonstrate the impact of active layer crystallinity on the accumulated charge and open-circuit voltage (Voc) in solar cells based on methylammonium lead triiodide (CH3NH3PbI3, MAPI). We show MAPI crystallinity can be systematically tailored by modulating the stoichiometry of the precursor mix, where small quantities of excess methylammonium iodide (MAI) improves crystallinity increasing device Voc by ~200 mV. Using in-situ differential charging and transient photovoltage measurements, charge density and charge carrier recombination lifetime are determined under operational conditions. Increased Voc is correlated to improved active layer crystallinity and a reduction in the density of trap states in MAPI. Photoluminescence spectroscopy shows that an increase in trap states correlates with faster carrier trapping and more non-radiative recombination pathways. We provide fundamental insights into the origin of Voc in perovskite photovoltaics and demonstrate why highly crystalline perovskite films are paramount for high-performance devices.