There are a whole lot of restrictions like this, because the society there is so small and isolated it basically doesn't have the infrastructure to deal with childbirth, very sick people and old people. But people who live there and have babies on the mainland do bring them back to Svalbard after they are born, and there is a kindergarten and a school. You're allowed to stay there after you are retired, but then you're on your own, you can't ask for any of the healthcare benefits that pensioners are entitled to.
There's also a whole lot of other exceptions, like different taxation from the mainland, exemption from needing a snowmobile licence to drive a snowmobile, permission to carry firearms almost everywhere, etc. The prices of e.g. milk and fresh produce are astronomical, while alcohol is cheap.
I'm aware of the death restriction due to the issue with it being permanently frozen there - the birth restriction mostly seems to be a simple lack of facilities that may lead to endangering the mother and child. I'd assume the birth restriction is more just down to liability and safety.
