Getting started with Socket.io in Swift on iOS involves setting up a Node.js server and installing dependencies using CocoaPods. The client-side app uses the Socket.io pod to connect to the server and receive real-time events, such as call status updates from Twilio's API. The app creates a data structure to represent phone calls and updates the table view when new or updated call status information is received from the server. To test the functionality, a Twilio phone number is used to make a call, which triggers the server to emit an event that the iOS app receives and updates the table accordingly.