The cell's cytoskeleton is like the bones in your body. They provide a framework for things to happen and allow the cell to keep its shape. Think how your organs would function without any bones to support them... Some of the cytoskeleton is used when when the DNA replicate as it provides a structure where the DNA can divide and physically split when a cell divides (amazing to watch this process). The cell even has special "walkers" that actually walk across the cytoskeleton to move things around. It is all very incredible.
