SlickNav looks to be a “Responsive Mobile Menu Plugin for jQuery”. Your theme probably ships with it. It’s written in JavaScript so try greping for the class name slicknav_menu in the JavaScript files included with the theme.
As an aside, another debugging tip for next time:
Most browser dev tools allow you to disable JavaScript. If you can’t find something in the PHP, try disabling JS on the page and reloading. If the $thing is not there anymore, it’s almost certainly due to JS.