Simplistically, you can think of a diode being a series connection of three elements("resistors") - one resistor for depletion region, and two resistors for quasi-neutral regions (in n- and p-type doped semiconductor).
Resistance in quasineutral region is much lower than the depletion region resistance at low applied voltages, that's why voltage drop on it, IR, is much smaller than a voltage drop on depletion region.
However, depletion region resistance is exponentially decreasing with applied voltage (simplistically, for diode I~exp(V/Vt) --> R=V/I ~ exp(-V/Vt), where Vt is a thermal voltage).
At high applied voltage (this corresponds to high injection level - i.e. when injected minority carrier concentration is higher than the majority carrier concentration), resistance of depletion region becomes lower than the series resistance of quasi-neutral regions, and so there is a significant voltage drop on quasineutral regions at high voltage / high injection level.
There are many good semiconductor device physics books that explain this, in particular - the "bible" by S.Sze, "Physics of semiconductor devices".