You can think of a private key like a really long, complicated password. Like, thousands of characters. But you don't have to type it in every time, you just let your phone store it for you and fill it in for you in apps (and now, websites).
To log into a website, your iPhone checks to see if your face is your face, and if it is then unlocks your private key to send it to the website. If it can't identify your face, then it won't enter your password.
Sending your actual facial data to a website would be bad because you can't change your face, so if you give your face to one site, then that site could use it to log in as you to other sites. But by just using it to unlock a private key, you (or apple on your behalf) can still change or de-authorize your private key, and use a unique one for each sites. Basically all the good practices you're supposed to follow when you use a password, and they aren't giving any site any special data that they could use anywhere other than their own site.