On an intuitive level you have 1 particles interacting with roughly n other particles at a distance inverse to the volume and those n other particles were interacted with by your first particle over a distance inverse to the volume this gives you (1*n)/(V*V) then consider that this kind of interaction is happening for all n particles not just 1 so n*(1*n)/(V*V) = (n/V)^2 is the winner. On a more in depth note...