Use an action hook to attach that enqueue to a specific page with conditional logic, similar to Chris' approach, just wrapped inside a function.
add_action( 'wp_enqueue_scripts', 'enqueue_custom_styles_or_scripts' );
function enqueue_custom_styles_or_scripts() {
// If it's not the front page, stop executing code, ie. return
if( !is_front_page() )
return;
// Else we reach here and perform the enqueue
wp_enqueue_style( 'your-style-handle', get_stylesheet_directory_uri() . '/yourfile.css' );
}
Note, despite the action name, it is also appropriate for styles to, i wish the devs would either rename that or add a style one alongside it, the naming would suggest it's for scripts only(but that's really not the case).
You can add the code to the top of your theme's functions.php file after the opening PHP tag(<?php) on a new line.