What is content.php file that is needed for Jetpack infinite scroll plugin?

There is no standard for a content.php in WordPress, that’s why you cannot find in the Codex. Some themes use it to render the post content.

So instead of …

while ( have_posts() )
{
    the_post();
    ?><li <?php post_class(); ?>>
    <?php
        the_title( '<h2><a href="' . get_permalink() . '">', '</a></h2>' );
        the_excerpt();
        wp_link_pages();
    ?></li>
    <?php
}

… they use …

while ( have_posts() )
{
    the_post();
    get_template_part( 'content', get_post_format() );
    ?></li>
    <?php
}

But you could use any name and it would still work. Jetpack follows a convention from WordPress’ last default themes here; these are using a content.php.