The diffusion rate increases as the surface area to volume ratio increases. The two are directly proportional.
Let's imagine one particle inside a cell that has to diffuse out of the cell. In a small cell with a high surface area to volume ratio, the particle has to travel a very small distance to diffuse out of the cell. In a large cell with a low surface area to volume ratio, the particle has to travel a very large distance to diffuse out of the cell. So, in the small cell the particle can diffuse more quickly because the particle has to travel less unit volume to get to the surface of the cell.
The important point is that the surface area to the volume ratio gets smaller as the cell gets larger. Thus, if the cell grows beyond a certain limit, not enough material will be able to cross the membrane fast enough to accommodate the increased cellular volume.
Level: Junior high