Why does a dissociation reaction shift to the right with the addition of an inert gas?
Dissociation obviously increases the number of moles.
The addition of an inert gas can affect the equilbrium, but only if the volume is allowed to change.

There are two cases on which equilibrium depends. These are:


Addition of an inert gas at constant volume:
When an inert gas is added to the system in equilibrium at constant volume, the total pressure will increase. But the concentrations of the products and reactants (i.e. ratio of their moles to the volume of the container) will not change.


Hence, when an inert gas is added to the system in equilibrium at constant volume there will be no effect on the equilibrium.


Addition of an inert gas at constant pressure:
When an inert gas is added to the system in equilibrium at constant pressure, then the total volume will increase. Hence, the number of moles per unit volume of various reactants and products will decrease. Hence, the equilibrium will shift towards the direction in which there is increase in number of moles of gases.


Consider the following reaction in equilibrium:
$$\ce{2 NH_3(g) ⇌ N_2 (g) + 3 H2 (g)}$$
The addition of an inert gas at constant pressure to the above reaction will shift the equilibrium towards the forward direction(shift to the right) because the number of moles of products is more than the number of moles of the reactants.

Please read about the Effect of adding an Inert Gas
