To integrate Nexmo Stitch's In-App Messaging into an iOS app using a UITableView, developers need to update their app layout, add a prototype cell, and configure the delegate and data source. They must then implement the required methods for UITableViewDataSource, such as numberOfRowsInSection and cellForRowAt, to display chat history. The implementation involves accessing properties from the ConversationClient instance, downcasting events, and assigning them to the prototype cell's textLabel.text property. By following these steps, developers can showcase a chat's history in their app using Nexmo Stitch In-App Messaging.