Public key is like your house address, everybody knows it, and knows it belongs to you. Private key is the actual key to your house, so you should never give it away.
Busy asks for posting keys because they will have to sign the posting transaction on the blockchain, to pair your article with your identity.
AFAIK, the posting key never really goes to Busy over the internet, it stays on your own computer. It is used by the JS library of Busy to sign the transaction locally and then the transaction is broadcasted to the blockchain.