The integration of Vonage's video calling with iOS Picture in Picture enables users to continue video calls uninterrupted while using other apps, enhancing multitasking and boosting productivity for professionals and collaborative work. To achieve this, developers need a Vonage API account, install CocoaPods, and use the Vonage Custom Video Rendering feature to process raw video data from Vonage's video renderer. The process involves creating a custom video render class that handles and processes incoming video stream data, converting YUV plane data into a format compatible with iOS's sample buffer, and implementing Picture in Picture component using AVPictureInPictureVideoCallViewController and AVPictureInPictureController.