Like others have said, the wiki claim of this preventing birthday attacks is wrong. Rather, this was meant to prevent against length extension attacks.
SHA-256(SHA-256(x)) was proposed by Ferguson and Schneier in their
excellent book "Practical Cryptography" (later updated by Ferguson,
Schneier, and Kohno and renamed "Cryptography Engineering") as a way
to make SHA-256 invulnerable to "length-extension" attack. They called
it "SHA-256d". We started using SHA-256d for everything when we
launched the Tahoe-LAFS project in 2006, on the principle that it is
hardly less efficient than SHA-256, and that it frees us from having
to reason about whether length-extension attacks are dangerous every
place that we use a hash function. I wouldn't be surprised if the
inventors of Bitcoin used it for similar reasons. Why not use SHA-256d
instead of SHA-256?
Note that the SHA-3 project required all candidates to have some
method of preventing length-extension attacks. Some of them use a
method that is rather like SHA-256d, i.e. they do an extra
"finalization" hash of their state at the end, before emitting a