How do I edit a plugin’s core properly?

If you look a little further down in the source of Breadcrumb Trail, you’ll notice this line:

/* Apply filters to the arguments. */
$args = apply_filters( 'breadcrumb_trail_args', $args );

The author has made it possible for you to do exactly what you need, without editing the core file! Simply hook into the breadcrumb_trail_args filter in your functions.php:

function wpse_104094_breadcrumb_trail_args( $args ) {
    $args['before'] = '';
    return $args;
}

add_filter( 'breadcrumb_trail_args', 'wpse_104094_breadcrumb_trail_args' );