The answer to how to add description into WordPress menu items is – easy :). All you need to do is extend walker class using nav_menu_item_title() filter which came with WordPress 4.4. So basically, all we need to do is to add $item->description where appropriate, which means inside the start_el() function which generates the menu link.
Have a look at the extended walker class, where I only added <span> element with the condition that if description does exist, <span> element along with description content will appear inside the <a> link element (for convenience, I removed all the comments from the walker class):
And this is how you would call this extension on targeted menu: