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.