You'll need custom coding to build a feature exactly like that into your store. If you do not have the necessary development experience, I would recommend hiring a developer to help you.

A simple way to do this, without any programming knowledge, is to create main categories for each letter of the alphabet, then to add all of your subcategories inside each of those letter categories. You will not get the same layout as that website, but it will be somewhat similar. The letters will all be listed vertically in the left hand column, then when you click a letter the subcategories appear directly underneath the letter.