The Paid By field was added to version 7.75.259 to track payments made in Student Centric systems and reduce data entry for Payer Centric Systems.
Student Centric: Since all payments should have the student under Payment From, there needed to be a field to determine who it was Paid By. You can search for records that exist in the system, or you can type in any name if a random 3rd party makes the payment and you do not want to enter them as a record.
Payer Centric: When a payment comes in from a party that isn't normally responsible for the charges, you can enter the normal Payer in the 'Payment From' field. You can track that 3rd party in the Paid By field. This prevents you from having to enter the 3rd party as a record in the system.
Note: You can also display the Paid By information on Statements. Select Configuration, Statement information, and 'Paid by' will be an option to display.