how to display an excerpt of the latest post on the homepage?

To show the excerpt from the latest post you can use query_posts.

Example Query Posts Showing the Latest Post with the Featured Image:

<?php query_posts('showposts=1'); ?>
<?php while (have_posts()) : the_post(); ?>
      <h3 class="home link"><a href="https://wordpress.stackexchange.com/questions/1575/<?php the_permalink(); ?>"><?php the_title(); ?></a></h3>

       <?php the_post_thumbnail(); ?>

        <?php the_excerpt(); ?>

        <?php endwhile; ?>

You will need to reset the query if you want to show content entered into your home page’s post editor.

<?php wp_reset_query(); ?>