Type systems can encode proofs of properties that you want your data to have. For example, suppose you want to write a function that returns the first element of a list.
head :: [a] -> a
if you feed this function an empty list, what will it do? You can reason from the type signature that the only thing it possibly can do is crash, since there's no way to produce a value of type a without knowing what a is in advance.
head (h:_rest) = h
head [] = error "Our function is badly behaved!"
data Nonempty a = Nonempty a [a]
head :: Nonempty a -> a
head (Nonempty x _rest) = x
Edit: I should probably add that "If it compiles it works" is absolutely not always true. It's more of a community in-joke than an actual belief. But it does turn out to be true surprisingly frequently.