It works like that. You print an empty contract from internet in two copies. Then you sign both copies, and you send it via mail to the bank. Bank signs both copies as well and sends one back to you. Then it provides the service.

Bank just offers you agreement to sign. Guy just offered the bank different agreement and the bank accepted it by signing.

Probably that's why in my country banks rather sends you already signed two copies of the agreement and you are supposed to sign both of them and send one back.