How does cell division solve the problems of cell growth?
One of the main methods of transport for substances within the cell is diffusion. Diffusion is the passive movement of substances along their concentration gradient: Particles move from an area with a high concentration to an area with a low concentration.
To see a reason why cells need to divide can be found in the law regarding the speed of diffusion: Fick's law (image below).
As can be seen in the image, the rate of diffusion is dependent on various factors. The factor we are looking for when we are talking about diffusion and cell size is the distance factor (D).
As you can see, the distance D has a negative impact on the diffusion rate as D becomes larger.
This means that cells that were to become really, really large would have a very slow transportation speed, to the point that the cell couldn't even function anymore.
This problem is solved by creating a new cell, which allows for utilization of more area by the same species as the mother cell.