I have some funds on the brave browser, but how do I extract the ethereum private key from the brave browser?
By design, you cannot do this.
At this stage in the project, fund transfer to the Brave browser is uni-directional. You can send funds in but you cannot take them out. The reason this works is because the idea is that you pay content creators for their work on the platform. The creators can withdraw the funds they receive, provided they create an account through Uphold.com. The reason for this is due to strict KYC and AML requirements in the United States.
With that said, this may change in the future. From /u/miyayes on Reddit:
The withdrawal feature will likely come when BAT Ads rolls around...which is when it would make sense anyway since that's when people will be earning $BAT in the browser.
This also helps people utilize the token as it is meant to be used, as opposed to speculating on the price.
As a final point, the browser warns you when depositing funds:
Reminder: The Brave Wallet is unidirectional and BAT flows to publisher sites. For more information about Brave Payments, please visit the FAQ.