The central phenomenon in the field of organic spintronics is the large magnetoresistance in thick organic spin valves. A prerequisite for understanding the magnetoresistance is a reliable description of the device resistance, or the I-V characteristics. Here I show that the observed I-V characteristics in the organic spin valves is incompatible with charge injection into the organic's lowest unoccupied molecular orbital or highest occupied molecular orbital but can be explained by electrons tunnelling into a broad impurity band located in the gap between these molecular orbitals. Voltage drop takes place mainly across depletion layers at the two electrode/organic interfaces, giving rise to electrode-limited charge transport. Spin-dependent electron tunnelling into the impurity band from the ferromagnetic electrodes results in spin accumulations inside the organic, which rapidly diffuses through the organic primarily via the exchange between impurity-band electrons. This picture explains the major magnetoresistance features and predicts enhanced capacitance in these devices.