Go to your inspection grid by selecting Inspections -> Scheduled Inspections. Choose your desired inspection and click on Actions -> Payments.
Click on the appropriate payment type (add Cash, Check, Money Order, etc.), enter the inspection fee collected, enter the check number if applicable and click on add payment. In order to run credit cards through the ISN you must set this up with your credit card vendor. (to enable it in your ISN select Settings -> Office Settings -> Payment Options)
After you have added a payment to an inspection the paid icon on your inspection grid will automatically change to a green square with white checkmark on the inspection grid.