I’m on iOS, I didn’t give them a Credit Card. I’m getting billed through Apple. I’d imagine the same thing happens on Android, PS4, or XBox One where a third party handles billing.
I signed up directly for the $9.99 no commercial option, so I can’t say for sure about what happens with the $5.99 option. But generally speaking this is how all these services work on iOS. I don’t think HBO, Hulu, or Starz has billing information on me.
They do all make you create an account and give them an email and password, which I thought was annoying, but the plus is, it does mean you can sign in on any device (i.e., I can signup on iOS, but then watch on my PS4 as well as my iPad).