Why to try get key out of white box crypto? How can one protect WBC itself?
This is an excellent question.  Unfortunately, I'm afraid I don't know of any really convincing answer.

In principle, one possible application of white-box cryptography would be to build a public-key cipher out of a symmetric-key cipher.  If you could build a white-box cryptographic implementation of AES encryption (say) where no one can recover the key nor figure out how to perform the decryption process, you'd have a public-key cipher.

However in practice I don't think that's actually the motivation for most work on white-box cryptography.  For instance, most published white-box crypto schemes can be easily inverted: given a white-box implementation of encryption, you can easily derive a white-box implementation of decryption.  Most published schemes focus on preventing you from recovering the key, instead of preventing you from figuring out how to perform the decryption process.  So, building a public-key cipher does not appear to be their motivation.

I don't know any really good answer to your question.  I know of two not-very-good answers:


One possible answer is that the code of the white-box cipher might be a lot larger than the key, and if a cracker wants to share it with others, they'll have to spend a lot more bandwidth sharing it than they would sharing the key.  However, this seems very weak and unconvincing.
Another possible answer is that white-box encodings are usually randomized: you can give a million people a randomized white-box implementation of AES decryption under key $K$ (the same key for everyone, but they each get a separately randomized white-box implementation).  Thus, if someone shares the code of their white-box decryptor, it uniquely identifies them.  However, in practice, this doesn't seem like a very effective defense against copyright infringement (prosecution is usually too expensive, and false identity too easy to forge).


So, I'm afraid I don't know of a strong practical motivation for white-box crypto.  Maybe someone else will have a better answer for you.