$wpdb query outputs php code instead of executing it

It’s very clear that your file is not executing PHP functions, and the problem probably is because you are saving this file as something.html, storing it inside your theme or WordPress installation directory.

What you can do, is to wrap this as a function or shortcode, and then create a page and use that shortcode or function inside it.

Here is a simple example to develop shortcode from your query:

add_shortcode('my-shortcode', 'create_my_shortcode');
function create_my_shortcode(){
    global $wpdb;
    $result = $wpdb -> get_results ( "
        SELECT 'VillageLeadersName' 
        FROM wp_villageleaderdb 
        WHERE VillageID = 'V001' AND VillageLeaderPosition = 'Sarpanch' LIMIT 0, 100
        " );
    return $result[0][1] . "<br />";
}

Put this code inside your theme’s functions.php file, and then head over to New > Page and create a page. Afterward, you can use this shortcode inside your page:

[my-shortcode]

This will execute and output your code. Also note that you can not echo content in a shortcode, you must return it.

Another approach is to simply wrap it as a function without creating a shortcode, and then call that function inside your templates.

function create_my_shortcode(){
    // Your code here
}

Now, make a copy of your theme’s page.php, rename it to page-whatever.php and modify it the way you like. After you finished editing, use create_my_shortcode() anywhere you want. In this method you can echo the content too.

Afterward, you can use this template while creating a new page in the back-end.

UPDATE

As stated in your comments, the results might be an array. To print the array, you can use a foreach like this:

$data="";
foreach( $result as $single_result ){
    $data .= $single_result;
}
return $data. $row['VillageLeadersName'] . "<br />";