Admin – Presentation – Theme Editor – category.php
Save a backup copy before you modify it, just in case you want to go back.
Locate "The Loop" it starts with this:
<?php while (have_posts()) : the_post(); ?>
try to understand the code, you can read it. You can delete the code for the_excerpt or the_content, also whatever you don't want like the_author, the_date, etc. The part you want to leave is basically like this
<h3><a href="<?php the_permalink() ?>" rel="bookmark" title="Permanent Link to <?php the_title(); ?>"><?php the_title(); ?></a></h3>
Notice the a href= which is making the Title a link to the full article.
Then leave alone the end of the loop and everything after
<?php endwhile; ?>