LR Circuit - qualitative look
From the circuit point of view, initially we have zero current so there is no VR (voltage drop across the resistor). Thus the full voltage of the battery is trying to change the current, hence VL = Vbattery, and so dI/dt = Vbattery /L.
However, as the current increases, there is more voltage drop across the resistor, VR, which reduces the voltage across the inductor (VL = Vbattery - VR), and hence reduces the rate of change of the current!