Once the payment has been acquired, the customer is shown their invoice or can be directed to another page on the site (for example, a page showing their order history).
They can also be sent an email, which contains a link to their invoice. (A copy of this email can also be sent to the site owner, to print out for their own accounts records - and for order fulfillment)
The URL of their invoice contains a hash, which acts as a substitute for login credentials. This means that a customer does not need to login to see their invoice. (However their privacy is preserved in that it is not possible to 'guess' the URL of any invoice even if the URLs of other invoices are known).