How to show post from category select

It looks like you are trying to model this on theme options code. You can’t do that. That is backend code. As written, this question is very broad– I am tempted to vote to close as “too broad”– but I will offer bare bones code to get you started:

echo '<form method="post">';
  wp_dropdown_categories();
  echo '<input type="submit" name="catsearch" value="Submit" />';
echo '</form>';
if (isset($_GET['cat']) && ctype_digit($_GET['cat'])) {
  $qry = new WP_Query( 
    array(
      'cat' => (int)$_GET['cat'] 
    )
  );
  var_dump($qry);
}

You will need to consult the documentation for wp_dropdown_categories and WP_Query, and will probably need to do a bit a research on HTML forms and basic PHP.