A signature is a mathematical mechanism that proves ownership. The blockchain will only transfer ownership of the Bitcoin if the signature is confirmed to be true. In the event that a false signature is used or no signature is given, the transaction is rejected. Blockchains are structured to use private keys to perform transactions. Simply put, only those with private keys can have their transactions recorded on blockchains. Furthermore, having a private key is said to be a way to maintain one's Bitcoin, but it is more accurate to say that private keys are the only way to show proof of ownership of one's Bitcoin.