Here is what worked for me.

Preethis answer almost got there, but no data was getting set. I needed to find a way of accessing the data in the array, and I used implode to do it.

So here is my final code that enabled me to create a search form that loops and displays categories, allows the user to do a keyword search via any number of categories, and display results IF the keyword exists in any of the categories selected:

I created a searchform.php file in my theme file and included the following HTML:

<form id="searchform" method="get" action="<?php bloginfo('url'); ?>">
<fieldset>

    <input type="text" name="s" value="" placeholder="Search..."></input>
    <button type="submit">Search</button>
    <div class="clear"></div>
    <div class="search_category_section">
    <?php
        $args = array('parent' => 0);
        $categories = get_categories($args);

        echo '<div class="search_category_section">';

        foreach ($categories as $category) {

        $thecatid = $category->cat_ID;
            echo '<div class="categorylist"><div class="parent_category_list_item"><input id="', $thecatid, '" type="checkbox" name="category_name[]" value="', $category->slug, '"><label for="', $thecatid, '">', $category->name, '</label></input></div>';
            $childcats=get_categories(array('parent' => $thecatid));
                foreach($childcats as $c) {
                echo '<div class="child_category_list_item"><input id="', $c->cat_ID, '" type="checkbox" name="category_name[]" value="', $c->slug, '"><label for="', $c->cat_ID, '">', $c->name, '</label></input></div>';
            }
        echo '</div>';
        }
    ?>
        </div>
</fieldset>

and then, my functions.php has the following code:

<?php
    function advanced_search_query($query) {
        if($query->is_search()) {
            $get_the_category_name = $_GET['category_name'];
            if (isset($get_the_category_name) && is_array($get_the_category_name)) {        
            $catnames = implode(",",$get_the_category_name);
            $query->set('category_name', $catnames);
            return $query;
            }
        }
    }
    add_action('pre_get_posts', 'advanced_search_query');
    ?>

if you do:

var_dump($get_the_category_name);

You’ll get:

array(2) { [0]=> string(22) "category-a" [1]=> string(25) "category-b" } 

Run implode() on that

string(48) "human-element-analysis,latent-defects-check-list" 

Stick that string in a variable, set it in query as category_name, win.