To add push notifications to an iOS app using Swift and Twilio Notify, developers need to create a Twilio Credential and provisioning profile for their Apple developer account, configure a Node.js server to authenticate the iOS app, create a Binding for Twilio to send notifications to the device, write code to register the device with APNS, receive notifications from APNS, and run the app on a real device using ngrok to make the Node server accessible. The process involves setting up an iOS app with a barebones project, registering the device with APNS, creating a Device Binding, and sending push notifications using Twilio Notify's Node.js server. With this setup, developers can send cross-platform notifications to their users, making it useful for various applications such as games, social media, and alerting users when new information is available.