Any time a cell divides, it makes more DNA for the new cell. So, when a starfish loses a leg for example, it creates copies of its DNA for each new cell that grows to replace it.

Any cell that exists makes itself according to the instructions of its DNA. So, many different genes are used to make new starfish parts. For example, when regenerating part of a digestive tract, the cells that are newly formed will use the genes that correspond to making a cell for the digestive tract.

Every cell contains the exact same DNA, but just different parts of the blueprint are used for each cell type. When the starfish first loses a limb, it responds by releasing growth hormones and what are called "heat shock proteins." These trigger the same genes that are used when the starfish is first developing, and it reads the DNA blueprint to decide what the new cells that form should differentiate into.