If you have a static page as your homepage then on the blog page you can display links to three categories above the blog posts, like this:
To set up the featured categories, open up the page you have assigned to display blog posts in the WordPress admin area.
On this page you will see the Featured Categories section. All you need to do is select the category you would like to display from the drop-down list.
For best results, make sure the categories you select have an image assigned to them. To learn how to assign an image to a category, check out this tutorial.