How do bitcoin addresses and private keys authorize payments?

"Bitcoin addresses appear on the public blockchain as strings of letters and numbers, and other people can use them to send funds to the associated balance. To make a payment, wallet software uses a private key uniquely associated with an address and mathematically combines it with the public information to create a valid authorization. The wallet selects sufficient funds from one or more addresses, sends the requested amount to the recipient, and can return any excess as change to an address controlled by the sender."