The process of regenerating new limbs (and even sometimes new bodies!) is very complex and only partly understood. Basically, regeneration is the ability of an animal's cells to make new body parts during adulthood, just like they did during embryonic development.
During embryonic development, most of an animal's cells take on a particular identity--they become blood cells, lung cells, bone cells, or whatever. This is called differentiation, and differentiated cells almost always keep their new identities forever. A special type of cell called a stem cell sometimes remains behind without a particular identity. Stem cells can be thought of as "permanently immature" cells that can decide later what fate to take on. (Stem cells in your skin are what make you able to heal after you've been cut or bruised.)

Starfish and some other animals have cells like this that can do more than just make new skin, they can differentiate into whole new arms. In order for them to do this, they need to receive the right signals from the rest of the body. If we lose a limb, we either don't send the right signals to our stem cells, or our stem cells are not able to differentiate into all the parts necessary for a whole new limb. Starfish seem to send the right signals, and their cells are able to differentiate properly, so they are able to regenerate whole new limbs.
We don't know yet exactly what those signals are, or why some stem cells are able to differentiate more than others. Scientists think that this may be easier for them, because their bodies are not as complex as ours.

Starfish, if you ever look closely at one, have a central region of the body from which the limbs arise. If a limb is severed, a new one (small) appears in the central region, and extends outward. I also know that starfish limbs can regenerate the central region (and thus all of the other limbs) as well.
If you're asking why can they do it but we can't, it's because of the program of development that starfish have. Their growth is indeterminate - they keep growing, and all cells retain the ability to grow into whatever proportion is needed. Our cells don't dothat. Why? That's an evolutionary question that is still being worked out, I expect.