What is the function of the "enable invoice" on a payment form?

I don't understand the function of the invoice....  If the form is a payment form, the user completes the payment in the form and receives a confirmation through email, which acts as the receipt.  

If I activate the invoice option, does the user receive an invoice that they pay from AFTER they submit the form?  Or does the invoice simply serve as a "receipt" for the payment they made online?

Where are payments to these sent invoices recorded?

Hello,

Yes, apart from the email submission, users can also get an invoice for their payment made. The invoice will contain the items purchased.

If you enable this option you can fill out the fields and also see a preview of the invoice. Users will receive the invoice in their email after they submit the form. 

The invoice is stored as a PDF in your JotForm account itself against that form. You can check that from your PDF Editor. When you click on Preview Invoice, it takes you directly to the PDF Editor linked to your form:

Here is an example of the invoice attached to the autoresponder or the notification email as required: