Quantum cryptography is pretty technical, but here is an elementary argument.
The Heisenberg uncertainty principle implies the no-cloning theorem for quantum-mechanical systems. If you could make an exact copy of a quantum-mechanical system, you could use one copy to measure its position, and the other to measure its momentum. This would violate the Heisenberg uncertainty principle. Therefore it should not be possible to do this kind of copying.
This doesn't immediately imply unbreakable codes, but it does imply something similar, which is a communications channel that can't be snooped on, because copying the stream of quantum information would require cloning.