DNA contains the blueprints for how to make the proteins and other molecules that make the cell run. Without it, cells would be unable to make the molecules they need to survive, and thus would die.
Cell replication is the process of going from one cell to two cells. Since the original single cell had only one set of DNA, it needs to copy it so that both resulting cells have the DNA they need to survive. Reproduction of multicellular organisms is essentially a lot of cell replications, followed by differentiation of those cells.
If DNA never replicated, meiosis(sexual reproduction) and mitosis(asexual reproduction) would slowly halve the size of the genome until each cell would die, which probably would not take long. Therefore, it is important that DNA doubles itself to account for the cells splitting during mitosis/meiosis. DNA replication is similar to RNA transcription.