How to Make a Categories Meta Box with Hierarchical Checkboxes on Frontend?

Try wp_terms_checklist() / wp_category_checklist. It will output a list of checkboxes named post_category.

You might need to include the source file too, because it’s defined within administration files.

Or use a custom walker:

class MyCategoryWalker extends Walker_Category{

  public function start_el(&$output, $term, $depth, $args){

    $args = wp_parse_args(array(
      'name'    => 'my_category_input',
      'checked' => array(),

    ), $args);

    extract($args);

    $checked = checked(in_array($term->term_id, $checked));

    ob_start(); ?>   

    <li>
      <input type="checkbox" <?php $checked; ?> id="category-<?php print $term->term_id; ?>" name="<?php print $name; ?>[]" value="<?php print $term->term_id; ?>" />
      <label for="category-<?php print $term->term_id; ?>">
        <?php print esc_attr($term->name); ?>
      </label>       

    <?php // closing LI is added inside end_el

    $output .= ob_get_clean();
  }
}

Use it like:

wp_list_categories(array(
  'walker'   => new MyCategoryWalker(),
  'name'     => 'my_category_input',       // name of the input
  'selected' => array(2, 5, 10),           // checked items (category IDs)
));