The Carrington Core framework aims to create maintainable WordPress websites by establishing conventions over code and using a template structure that makes functionality and features more obvious. It follows WordPress' core structure and uses context-based templates located in different directories to display content for posts, pages, comments, and general sections. Templates can be further customized based on additional context like author, category, tag, or post format. The framework is intended to make WordPress theme development more accessible to front-end developers.