Why html input submit not assigning $_POST array value

It's because by this condition if (isset($_POST['id_number_submit'])) ! When you click save_changes_submit submit button ,there is no contain id_number_submit in POST data.So its doesn't not run your condition.The solution to change your condition

echo '<form action="" method="post">ID Number:<input type="number" name="id_number" value="number"><br><br><input type="submit" value="Search Product" name="id_number_submit">';

if (isset($_POST['id_number_submit'])) {
    $id_number = $_POST['id_number'];

    $q = mysqli_query($conn, "SELECT * FROM product WHERE id='$id_number'");

    $row = mysqli_fetch_array($q);

    if (empty($row)) {
        echo "Error: Invalid product id";
    }
    else{

    $product_name = $row['product_name'];
    $product_des = $row['product_des'];
    $product_price = $row['product_price'];


echo '<br><br><input type="number" name="product_price" value="'.$product_price.'">';
echo '<br><br><input type="text" name="product_name" value="'.$product_name.'">';
echo '<br><br><textarea rows="8" cols="100" name="product_des">'.$product_des.'</textarea><br><input type="submit" value="Save Changes" name="save_changes_submit">';    

   }
}
if (isset($_POST['save_changes_submit'])) {
    $c_product_price = $_POST['product_price'];
    $c_product_name = $_POST['product_name'];
    $c_product_des = $_POST['product_des'];
    echo $c_product_price;
    echo $c_product_name;
    echo $c_product_des;
}

echo '</form>';

Because you didn't assign which php file to handle with the form .Change the action of form like this:

echo '<form action="<?php echo $_SERVER['PHP_SELF'];?>" method="post">ID Number:<input type="number" name="id_number" value="number"><br><br><input type="submit" value="Search Product" name="id_number_submit">';