When your wallet pops up a “Please sign” window, what exactly are you signing?
My understanding: A signature requested by your wallet isn’t necessarily for a transfer; it could also be a token approval or an off-chain signature. The consequences of signing may be similar to making a direct transfer.
Digital signatures use a pair of keys: the private key generates the signature, and the public key verifies it. You can’t derive the private key from the public key.
When transferring $BTC , the wallet uses your private key to sign the transaction, and nodes across the network use the public key to verify the signature. Current price: 82045.15.

For reference only. Use at your own risk.