When you create an expense entry from a vendor bill or a reimbursable expense entry for an employee, it shows up for payment in the Bill Payments screen. When you make a payment for the bill (or reimbursable expense) there, the linked expense entries are automatically marked as Paid. You can also mark expense entries as paid manually by updating their Paid flag in the Expense Entries > Details > Additional screen.
Note: As the Paid flag only makes sense for reimbursable expenses and vendor bills, if you check it for non-reimbursable expenses, CORE automatically checks their Reimbursable flag to indicate they are supposed to be reimbursable expenses. Check CORE Help Center for details on the Paid flag.