AWS Lambda@Edge provides a way to run custom code for each CloudFront request, allowing for more stable deployment processes and the ability to slowly roll out new versions of an application like Link to end users. The approach involves capturing requests to a specific URL, implementing custom logic to select which version is served, and then switching traffic to the new version after a certain period if no issues arise. To ensure consistency in serving different versions to users, the system uses IP address and User-Agent string hashing to assign users to either the old or new version. The deployment process has been successful for over 30 releases, but it's also important to set up proper metrics to monitor during the rollout period to catch any issues that may arise.