When publishing a post on Contentful, it allows the site to rebuild automatically by triggering new builds when content updates through Netlify's webhooks, which accept HTTP POST requests. To set this up, configuration is required within both Contentful and Netlify, with steps including navigating to Build Hooks in Netlify Settings, adding a webhook in Contentful Settings, and selecting the endpoint URL from Netlify. Once configured, publishing or editing posts in Contentful will trigger a new build in the Netlify dashboard, pulling in content and publishing it to the site, providing an automatic deploy feature for the CMS.