Add a columnar category browser in a new tab below the template search form (next to the existing Favorites tab).
This will show a column-based browser for the category hierarchy, starting at a designated root category such as Category:Templates. Each column will list both templates and subcategories, in their sortkey order. Clicking on a template will load the template form, and clicking on a category will open a new column with that category's members.
A rather rough mock-up:
The UI will be something similar to Apple's column view.
This task is about browsing templates and their categories. For searching, see T392245: Template discovery: search templates by category.
