To record a payment in Sprout Studio that was received outside of Sprout Studio, it's as simple as:
Going into the order
Scrolling down to the invoice and clicking the invoice name
Clicking the "Pay" button in the "Manage Payments" Window
When adding in the payment, you can specify how it was paid, when it was paid, and how much was paid. You can also add in any additional payment notes that may have come up when your client paid (such as a transaction number, etc.).
When you're done, click either the "Add Payment + Email Receipt" button to add the payment to the order and email the receipt, or click the "Add Payment (No Receipt)" button from the arrow next to the previous button.
Please note, this method is only for recording payments already received. It will not charge your clients through your Payment Processor.