How come cells don't just continue to grow larger as organisms grow larger? Why do cells divide?

It needs to maintain a high SA:V ratio.

If you think about it, it makes sense. The has to move nutrients inside the cell, and move waste outside it. It would be very difficult to do this with a huge cell, since there is too much stuff inside the cell(volume), compared to the relatively low amount of cell membrane.

Surface Area increases slower than Volume, since it is a squared function and not cubed. Therefore, a cell has to be small, by dividing many times, so that the cell membrane is large enough to channel things in and out of the cell efficiently.