image in post not shown on home

The page template your theme is using to display your front page (home.php – if you don’t have that one then it is index.php) is using this tag –> <?php the_excerpt(); ?> instead of <?php the_content(); ?>

When you use <?php the_excerpt(); ?> WordPress strips out all formatting. If you want the front page to look like individual post pages change that tag.

More reading on <?php the_excerpt(); ?> and <?php the_content(); ?> here.