The Postmark API provides an easy way to retrieve bounced email addresses. The API can be accessed through the `RestClient` gem in Ruby, which allows developers to execute HTTP GET requests and parse JSON responses. By setting the correct parameters and headers, including the `X-Postmark-Server-Token`, users can obtain a list of bounced emails with various fields such as email address, bounce type, and inactive status. The response can be easily parsed into a usable data format using Ruby's standard library. Additionally, Postmark provides an official Ruby gem that offers a more convenient interface for accessing bounce information. By leveraging the Bounces API, developers can create their own reporting and analytics tools to improve email deliverability.