You need to state the action of the form as the name of the file. So, if the file you are working on is name example.php it should look like so:

$email = trim(get_variable_value('email'));
$code = trim(get_variable_value('code'));
$code2 = trim(get_variable_value('code2'));

//make sure to validate the input!! if it's not valid send back

<? if(!isset($email)){ //alternatively, you can add a hidden input `stage`. I just don't like it, as it can be bypassed.
?>
    <form name="transferToNewForm" action="example.php" method="post">
        <label for="email">Email Address:</label> <input type="text" id="email" name="email">
        <input type="hidden" id="code" name="code" value="42">
        <input type="submit" value="Submit1">
    </form>
<? } 
if(isset($email) && isset($code) && $code == 42){ ?>
    <form name="transferToNewerForm" action="example.php" method="post">
        <label for="email">Email Address:</label> <input type="text" id="email" name="email" value="<?= $email ?>" readonly>
        <input type="number" id="code2" name="code2">
        <input type="submit" value="Submit2">
    </form>
<? } 
if(isset($email) && isset($code2) && $code2 == 42){ ?>
    <form name="finalFrom" action="theActualPageYouWant.php" method="post">
        <label for="email">Email Address:</label> <input type="text" id="email" name="email" value="<?= $email ?>" readonly>
        <input type="number" id="code2" name="code2">
        <input type="submit" value="Submit3">
    </form>
<? } ?>

You could try this approach :

<?php 
function getFormContent() {
  $email = $_POST['email'] ?? null; //$email equals to null if it is undefined, to prevent errors

  if (!$email) {
    echo '<input type="text" name="email" placeholder="Email.." />';
    echo '<input type="hidden" id="code" name="code" value="42">';
  } else {
    echo '<input type="hidden" name="email" value="'.$email.'" />'; //keep email set after submitting first step
  }

  if ($email && $_POST['code']) {
    if (isset($_POST['code2']) && $_POST['code2'] == $_POST['code']) {
      //final form
    } 
    else { //second step
      echo '<input type="number" id="code2" name="code2">';
      echo '<input type="hidden" id="code" name="code" value="42">'; //we still need this in case user submitted a wrong code
    }
  }
}
?>

<form method="post">
  <?= getFormContent() ?>
  <input type="submit" />
</form>