You need to use both the [link-library-cats] and [link-library] shortcodes to get the result you are looking for.
The [link-library-cats] shortcode is used to display a list of categories for your links, while the [link-library] shortcode will display the links themselves.
Do you want the categories to be displayed first, followed by a list of all categories with their links? Or do you want to only see links from a first category, then change the category by clicking on a category name? Both of these behaviours and more are available through configuration options.
Let me know if this helps getting you closer to your goal or if you have further questions.