What is the purpose of the "Links" field in the "Manage Display" UI?
The Links section is used normally in Teaser section/display to display links like add comment, read more, etc.

On Node Full Display, it is depending on configurations we do. Such as "Add new comment" link will only be visible when we disable "Show reply form on the same page as comments" check-box in comment field settings. You can also give a try to Flag Module or other contributed modules which adds links on Node display.

Please check screenshot below (Teaser Display):


The above screenshot is of default Teaser Display of node type article, which is being displayed on site home page. When we hide Links from teaser display, these links are no more visible in node teaser.

Following screenshot shows links working in Node Full Display:
Depending on configuration we do in comment field (check screenshot below), comment link will be visible on Full Display.


Comment link visible, when checkbox is disabled.


Comment link is not visible, when checkbox is enabled.