The reason is because the Minimal Download Strategy feature does not download all the necessary scripts to enable the "Connect to Outlook" feature.
What is Minimal Download Strategy? It is a new feature in SharePoint 2013 that improves client rendering performance and fluidity when navigating from page to page by download only the changes between two compatible pages. Fewer bytes will be downloaded and the page will appear more quickly.
To enable the "Connect to Outlook" button, you can disable the Minimal Download Strategy, but this is not recommended as you will lose out more of the cool capabilities that it can perform.
The recommendations to get the "Connect to Outlook" button, just refresh the page by pressing the "F5" button or refresh on the browser, the button will be enabled.
God bless I was wondering why it was grayed out~!