Refresh stránky odešle $_POST znovu.

<?php 
session_start();

var_dump($_SESSION);
?>

<h4>Nastavení hesla</h4>
<form action='generuj_hesla.php' method="post">
    Síla hesla: <select name="sila" style="margin-bottom: 10px;">
        <?php
       for ($i = 4;$i <= 20; $i+=2) {
    echo "<option value='$i'>".$i."</option>";
}
?>
</select><br/>
Počet hesel: <select name="pocetHesel" style="margin-bottom: 10px;">
    <?php
   for ($i = 4;$i <= 20; $i+=2) {
    echo "<option value='$i'>".$i."</option>";
}
?>
</select><br/>
S českými znaky: <input name="czZnaky" value="true" type="checkbox" style="margin-bottom: 10px;"><br/>
<input type="submit" name="odeslano" value="Odeslat" style="margin-bottom: 10px;" /></form> 


<div id="hesla">
    <?php
    if (isset($_SESSION['hesla']) && count($_SESSION['hesla']) > 0) {
    ?>
                        <ul>
                            <?php
            foreach ($_SESSION['hesla'] as $heslo) {
            ?>
                                    <li>
                                                                <?php echo $heslo; ?>
                                                                                        </li>
                                                                        <?php
                    }
                $_SESSION['hesla'] = array();
                ?>
                                    </ul>
                                    <?php
    }
    ?>
                    </div>

<?php 
session_start();

function pridej_heslo($heslo)
{
  if (!isset($_SESSION['hesla']))
    {
      $_SESSION['hesla'] = array($heslo);
    }
  else
    {
      array_push($_SESSION['hesla'], $heslo);
    }
}

function Pass_Gen($sila, $pocetHesel, $czZnaky = false)
{
  $cisla = range(0, 9);
  $mala = range("a", "z");
  $velka = range("A", "Z");
  $ceskeZnaky =
    array ("á", "č", "ď", "ě", "é", "í", "ň", "ó", "ř", "š", "ť",
       "ů", "ú", "ý", "ž", "Á", "Č", "Ď", "É", "Í", "Ň", "Ó",
       "Ř", "Š", "Ť", "Ú", "Ý", "Ž");
  if ($czZnaky)
    {
      $znaky = array_merge($cisla, $mala, $velka, $ceskeZnaky);
    }
  else
    {
      $znaky = array_merge($cisla, $mala, $velka);
    }

  for ($i = 0; $i < $pocetHesel; $i++)
    {
      shuffle($znaky);
      pridej_heslo( implode("", array_slice($znaky, 0, $sila)));
    }
}

if (isset($_POST["odeslano"]))
  {
    if ($_POST['sila'] != "" && $_POST['pocetHesel'] != "" && empty($_POST["czZnaky"]))
      {
        Pass_Gen ($_POST['sila'], $_POST['pocetHesel']);
      }
    elseif ($_POST['sila'] != "" && $_POST['pocetHesel'] != ""
        && !empty ($_POST["czZnaky"]))
    {
      Pass_Gen ($_POST['sila'], $_POST['pocetHesel'], $_POST["czZnaky"]);
    }
    else
    {
      // tady je problem, asi by byla treba dalsi promenna v SESSION, ktera bude obsahovat chybove hlasky
    }
  }
// nakonec presmeruj, tady nic nevypisuj!
header ("Location: generator_hesel.php");