Braintree + Wordpress: How do I associate new users with a BT customer ID?

I am new to web development. I decided to start by using WordPress.
I am reading through the BT API and I'm at the section to use the Customer object to create, update, find, and delete a customer/payment info.

But I don't understand the correlation between WordPress and integrating payment processing yet. In WordPress, users create a login specifically for my site, right?

Would I need to hack WP to allow them to also associate a Braintree customer ID with that login? I don't understand the relationship between WordPress and integrating payment processing....