Why does an appointment display as "Waiting for Client Approval"?

This means that the service booked has Confirmation Required turned ON - This means, when a client goes to book that service, they are submitting a request to be able to book the appointment. When the appointment request is edited, it will ask if you would like to send a confirmation to the client to review, which is why it says "waiting for client approval"

If you do not want this, you can 1) turn off Confirmation Required on the service, or 2) approve the appointment instead of submitting it to the client for approval.