There are a few different ways clients can prepay for your services/classes via Full Slate.
When "Paid via CC $xx" is displayed when you click on the appointment or class, this means that the client paid in advance via the credit card processor you are integrated with for their appointment or class. You would need to check with your payment processor for the specifics on each transaction. You can determine what processor you are integrated with under Setup > Client Payments.
If you see "Paid $xx" this means that the client prepaid with their voucher (ie: a previously purchased gift certificate or package). You can check on the Sales tab for details on when the client prepaid this amount. For more on Gifts & Packages, please see here.
If you have any questions, feel free to reach out via the "Support" link in your account.