CallKit is used to handle VoIP push notifications sent to an iOS device when using the Vonage Client SDK for iOS, allowing integration of the app into the system so that it can look like a native iOS phone call. The process involves setting up push certificates, adding push notification capability, generating a push certificate, uploading the certificate to Vonage's API dashboard, and registering a device for push notifications. Once registered, the client SDK handles incoming push notifications by implementing the CXProviderDelegate protocol. The ClientManager class is used to interface with the Client SDK and handle user actions on the CallKit UI.