You can work through the derivation, but I think you are after a more intuitive answer to the question. Here is the way I think about it.
You will know that the I-V curve a resistor is $V=IR$. That is to say that the when you put a voltage across a resistor the current is linearly related to the voltage simply through a constant of proportionality, R, the resistance. Let's think why that is, before we tackle the pn-junction. Think of the voltage you are applying to the resistor as a pressure and think of the resistor as a pipe. The more pressure you apply, the fast water will exit the end of the pipe. That is to say, the move voltage you apply to a resistor the higher the current. In real materials electrons will scatter off atoms and impurities so the material has an intrinsic resistance. The scattering is the only mechanism which slows down the drift of electrons. However, in a semiconductor diode there is a second mechanism which prevents the carriers from drifting along in the direction of the applied electric field: the built-in field of the pn-junction.
Unlike the resistor where the electrons see a flat 'landscape' throughout the material, in a diode the electrons see a flat area on the n-side but see the p-side as a raised plateau. To get through the material the electrons must first roll up this potential gradient (i.e. pass through the junction region).
Electrons have random thermal motion, similar to the motion of gas molecules in the air (electrons are Fermions, they obey Fermi statistics but the comparison with classical particles holds. In fact it is common to think and mathematically treat electrons as a classical 'electron-gas'). Some electrons will randomly acquire enough thermal energy to roll up the hill and reach the plateau. The (Fermi-Dirac) function describes the distribution of electrons "vertically" above the minimum energy: there are many electrons at "ground level" which reduces exponentially has you go higher in energy. This means that for every electron that acquires enough thermal energy to roll up the hill, there are exponentially more that do not.
When you increase the (forward) bias on the diode, the hill begins to become flatter. Because the electrons are distributed exponentially in height above the our imaginary landscape, it means that a linear change in the voltage causes an exponential change in current.
Semiconductor have defects; region of the material where there is a foreign atom or the lack of an atom. If an electron comes close to a defect then one of two things can happen:
1) If the defect is the type that can also trap holes then when the electron is trapped, the it will recombine with the hole. That is to say the current has been reduced because of non-radiative recombination.
2) If the defect can only trap electrons then the electron will be trapped at the defect site until is randomly acquires enough thermal energy to escape.
Either way, the effect is the same: the current is reduced. $n=1$ is an ideal diode (only radiative recombination), $n>1$ non-ideal diode (non-radiative recombination).
So that is my hand waving tutorial on the diode equation.