How do you create a “virtual” page in WordPress

There are two types of rewrite rules in WordPress: internal rules (stored in the database and parsed by WP::parse_request()), and external rules (stored in .htaccess and parsed by Apache). You can choose either way, depending on how much of WordPress you need in your called file.

External Rules:

The external rule is the easiest to set up and to follow. It will execute my-api.php in your plugin directory, without loading anything from WordPress.

add_action( 'init', 'wpse9870_init_external' );
function wpse9870_init_external()
{
    global $wp_rewrite;
    $plugin_url = plugins_url( 'my-api.php', __FILE__ );
    $plugin_url = substr( $plugin_url, strlen( home_url() ) + 1 );
    // The pattern is prefixed with '^'
    // The substitution is prefixed with the "home root", at least a "https://wordpress.stackexchange.com/"
    // This is equivalent to appending it to `non_wp_rules`
    $wp_rewrite->add_external_rule( 'my-api.php$', $plugin_url );
}

Internal Rules:

The internal rule requires some more work: first we add a rewrite rule that adds a query vars, then we make this query var public, and then we need to check for the existence of this query var to pass the control to our plugin file. By the time we do this, the usual WordPress initialization will have happened (we break away right before the regular post query).

add_action( 'init', 'wpse9870_init_internal' );
function wpse9870_init_internal()
{
    add_rewrite_rule( 'my-api.php$', 'index.php?wpse9870_api=1', 'top' );
}

add_filter( 'query_vars', 'wpse9870_query_vars' );
function wpse9870_query_vars( $query_vars )
{
    $query_vars[] = 'wpse9870_api';
    return $query_vars;
}

add_action( 'parse_request', 'wpse9870_parse_request' );
function wpse9870_parse_request( &$wp )
{
    if ( array_key_exists( 'wpse9870_api', $wp->query_vars ) ) {
        include 'my-api.php';
        exit();
    }
    return;
}