The theme-compat directory is comprised of a set of deprecated (since WordPress 3.0) files that WordPress used to use as fallback template-part files, in case the active Theme failed to include them.
In other words, if a Theme used the get_header() template tag, but failed to include a header.php template-part file, WordPress would fall back to /wp-includes/theme-compat/header.php.
The template-part files in theme_compat are deprecated, unmaintained, incredibly old (by core/current-Theme standards), and should not be relied upon by any current Theme.