Why does Area of Interface depends on size of particles?
Area of interface is the area of contact between two phases. If one phase is dispersed in another one, it then consists in the sum of the surface for each particle/droplet.

If you consider a spherical particle, its surface is $S=4\pi r^2$ and its volume $V=\frac{4}{3}\pi r^3$. Now consider the ratio $\frac{S}{V}=\frac{3}{r}$, so when the size of a sphere (but it is true for any shape) diminishes, the ratio $\frac{S}{V}$ increases. In other words, if you take an initial volume that you divide in small spheres, the global volume does not change (the sum of all spheres) but the surface increases.