Youtube never knows the private half of your key pair. That never leaves your system.
Anything encrypted with the private half can only be decrypted with the public half, and anything encrypted with the public half can only be decrypted with the private half. These halves are known as the public key and the private key. Each side of the connection generates their own key pairs.
We both generate a set of keys, and exchange the public halves with each other. I then want to send you a message:
I first encrypt it using my private key, I then encrypt it again using your public key and send that to you.
In order to read that message, you first decrypt it using your private key. This ensures the message was intended for you and wasn't modified in transit, as you are the only one with access to that private key and only its matching public key could have been used to encrypt that layer.
You then decrypt it a second time using my public key. As I'm the only one with access to my own private key, you can be sure the message was sent by me.
As long as that resulted in a readable message; You've now verified who sent the message, that it was intended for you, and that the contents have not been modified or read in transit.
All this, including the key exchange is handled for you by the https (tls) protocol every time you connect to a website. Each of the messages sent between you and the site are encrypted in this manner.