They can't. AI generated code cannot be copyrighted. They've stated that claude code is built with claude code. You can take this and start your own claude code project now if you like. There's zero copyright protection on this.
There's no major lawsuits about this yet, the general consensus is that even under current regulations it's in the grey. And even if you turn out to be right, and let's say 99% of this code is AI-generated, you're still breaking the law by using the other 1%, and good luck proving in court what parts of their code were human written and what weren't (especially when being sued by the company that literally has the LLM logs).