How to add html to specific section

I am trying to style my dropdown menu similar to the one at https://www.dmv.org, it is a menu wide dropdown with links and various content. I am still in the learning stages of using child themes, I understand the CSS part but I don’t know how to add my own html to the php child part. If you hover over the “Insurance” menu item on the dmv.org site you will see what I am trying to accomplish. I can modify an existing dropdown using css to match it exactly, but I can’t figure out how add the content. I can do it in firebug by using “Edit HTML” but i don’t know how to translate that into my child theme. Thanks in advance for helping a newbie!