The React Native Video Call App uses Call Keep for push notifications on iOS devices, allowing users to initiate video calls and receive notifications in the background. The app is built using React Native and Firebase, with a focus on server-side setup using Node.js and the `node-apn` library. The client-side code sets up the necessary libraries and handles push notification registration, while the server-side API sends APN notifications for iOS devices and FCM notifications for Android devices. With this implementation, users can now make video calls on both Android and iOS devices, with features like screen sharing and chat messaging planned for future development.