How can I make […] an active link

Here’s how you can do that. Add the following to your theme’s functions.php template file:

remove_filter('get_the_excerpt', 'wp_trim_excerpt');
add_filter('get_the_excerpt', 'my_custom_excerpt');

function my_custom_excerpt($text) { // Fakes an excerpt if needed
        if ( '' == $text ) {
                $text = get_the_content('');
                $text = apply_filters('the_content', $text);
                $text = str_replace(']]>', ']]>', $text);
                $text = strip_tags($text);
                $excerpt_length = 55;
                $words = explode(' ', $text, $excerpt_length + 1);
                if (count($words) > $excerpt_length) {
                        array_pop($words);
                        array_push($words, '[<a href="'. get_permalink() . '">...</a>]');
                        $text = implode(' ', $words);
                }
        }
        return $text;
}

All I’ve done is remove the wp_trim_excerpt function from the get_the_excerpt callback, and then modified wp_trim_excerpt into what I’ve called “my_custom_excerpt“, just adding the link to the post around the ellipses.