After looking at the widget source, I've deciphered what @charvan was getting at.
The problem is not with the widget per se, but with the way that other themes include widgets within their rendered pages. For the plugin to work, it needs its widget code to be contained within a div of class
simple-social-icons. My theme wasn't applying class names properly to each widget, meaning that the plugin is not styled properly; it relies on being assigned a specific font which includes the social media icons.
I'm using a theme called fullscreen. To fix the problem, I went into the wp-content/themes/fullscreen directory, and looked for where widgets are included (at the command line, "grep -r before_widget ."). It turns out they're in lib/widgets.php.
Beforehand, the relevant line of code looked like this (hoping formatting works OK):
'before_widget' => '<div class="item">',
To fix it, I amended it to:
'before_widget' => '<div class="item %2$s">',
(I had to make this change in 3 different places to amend the way that widgets are displayed in different parts of the page.)
Hope that helps.