Well, there are several ways to do it. The easiest (but not the smartest) is to use the default loop of the index page but add the condition that will add month titles. To do it place the following code right after
<?php while (have_posts()) : the_post(); ?>
in the index.php of your theme
Code:
<?php
if($current_month != get_the_time('m')){
?>
<h1><?php the_time('F') ?></h1>
<?php
}
$current_month = get_the_time('m');
?>