That’s because the code you listed only includes a reference for the title and the link. Here’s your original code, annotated …

Original

<?php $debut = 0; //The first article to be displayed ?>

<?php while(have_posts()) : the_post(); ?>

<!-- Display the title of the current page that's listing your posts -->
<h2><?php the_title(); ?></h2>

<!-- Create an unordered list to display a list of posts -->
<ul>

    <!-- Get a list of all posts, not including the first post -->
    <?php $myposts = get_posts('numberposts=-1&offset=$debut');

    // Loop through each post that your just grabbed
    foreach($myposts as $post) : ?>

        <!-- Add a list element for the post -->
        <li><?php the_time('d/m/y') ?>: <a href="https://wordpress.stackexchange.com/questions/51068/<?php the_permalink(); ?>"><?php the_title(); ?></a></li>
    <?php endforeach; ?>
</ul>

<?php endwhile; ?>

This will create a block that looks something like this:

<ul>
    <li>5/2/2012: <a href="">Post title</a></li>
    <li>5/2/2012: <a href="">Post title</a></li>
</ul>

What You Should Do Instead

You just need to update your post template. Right now, it’s grabbing the posts and outputting the date published, a link, and the title. So replace everything inside the <ul> ... </ul> with this:

<div class="post-list">
    <?php $myposts = get_posts( 'numberposts=-1&offset=$debut' );
    foreach( $myposts as $post) : setup_postdata( $post ) ?>
        <h1><?php the_title(); ?></h1>

        <!-- Only display part of the post so the user has to click "More!" -->
        <?php the_excerpt(); ?>

        <a href="https://wordpress.stackexchange.com/questions/51068/<?php the_permalink(); ?>">More!</a>
    <?php endforeach; ?>
</div>