Looking into the Theme Customizer API, it appears that theme modifications are stored as theme_mods which can be access using get_theme_mod( $mod_name ) or (if you want to grab them all) get_theme_mods().
Peeking into the source of get_theme_mods(), I see that the mods are stored in the site’s options table as theme_mod_$mod_name. So the theme modifications are stored in the database, not in any files.