Connecting MongoDB Stitch to Google Places leverages Stitch functions, which are written in JavaScript ES6 and can be used for coordinating data access or doing light processing alongside a query or insert. Stitch provides an HTTP service that can communicate with external services like Google Places, enabling applications to retrieve contextual information about businesses without maintaining the data themselves. Stitch Triggers enable reactivity to database inserts, updates, deletes, and replaces, allowing functions to be executed automatically in response to changes. By integrating MongoDB with Google Places using Stitch, developers can build intelligent apps that provide users with valuable information without requiring user input or maintenance of the data.